I don't know what will be revealed in future programs about any extra lives the Doctor may have had. I have a little evidence and a theory.
I don't know what the answer is.
But here is some more evidence.
In the Fourth Doctor series from the 13th season "The Brain of Morbius", 3 to 24 January 1976, written by Terrance Dicks and Robert Holmes there is a time lord duel between the Doctor and the brain of the evil Time Lord Morbius (installed in a monstrous body). As they mentally battle each other and attempt to force their minds back to the time of birth and then before birth, and thus into non existence, a monitor shows faces.
It shows the face of the 4th (Tom Baker) Doctor, then the face of the 3rd (Jon Pertwee) Doctor, then the face of the 2nd (Patrick Troughton) Doctor, then the face of the first, (William Hartnell) Doctor, and then it shows other faces. Morbius says "How far Doctor? How long have you lived?".
After about eight, if I remember correctly, more faces are seen after Hartnell's, Morbious's mind is killed and his mindless monster body goes on a rampage until it is physically killed.
And fans assumed that the eight faces seen after the face of William Hartnell were faces of some or all of the Doctor's incarnations before the William Hartnell Doctor, and thus that the Tom Backer was the 12th or later incarnation of the Doctor.
But in the next season, the serial "the Deadly Assassin" 30 October to 20 November 1976, written by Robert Holmes, featured the almost dead Time Lord The Master returning to Gallifrey. The Master had used up all his regenerations (by being fatally injured or wounded in various conflicts, I guess) but sought to gain new life on Gallifrey. A Time Lord said that a Time Lord gets twelve and only twelve regenerations.
It is possible he meant that that a time Lord gets twelve incarnations, which would put the Fourth Doctor in his last regeneration. But it is much more likely that the Time Lord meant that Time Lords get 12 regenerations from one incarnation to the next, and thus that a Time Lord would have thirteen incarnations or lives, meaning that the Fourth Doctor would have one more regeneration into his thirteenth incarnation or body.
Since either possibility would limit the number of future bodies the Doctor could have, and thus the number of future actors who could portray him, fans were not happy and so most fans decided that the eight unknown faces seen in "The Brain of Morbius" were not all previous incarnations of the Doctor. Some or all of the faces should have been those of the previous Time Lord lives of Morbius.
It was established in The Deadly Assassin (1976) that a Time Lord can regenerate twelve times before permanently dying – a total of thirteen incarnations. The 20th Anniversary special, the 1996 television film and the 2013 special "The Time of the Doctor" all confirm this with the latter showing that the Time Lords can circumvent the cap of 12 regenerations in total by giving a Time Lord extra regenerations.
So the accepted theory is that the William Hartnell Doctor, seen in the first few seasons, was the first incarnation of the Decctor, and that all of the eight unknown faces in "The Brain of Morbius" were lives of Morbius.
And that is supported by once or two references to the William Hartnell Doctor as "the first" in stories like "The Three Doctors" 30 December 1972 to 30 January 1973, and "the Five Doctors" special 23 November 1983, written by Terrance Dicks. And possibly various Doctors in the "new Who" shows since 2005 have been referred to by numbers consistent with the William Hartnell Doctor being the first incarnation.
In The Brain of Morbius (produced shortly before The Deadly Assassin), a sequence of faces displayed during a mental battle between the Fourth Doctor and Morbius imply that the Doctor had at least eight incarnations prior to the First Doctor. This is contradicted by multiple dialogue references throughout the programme (particularly in The Three Doctors, Mawdryn Undead and The Five Doctors) as well as the fact that the Doctor has regenerated nine times since then (as confirmed in "The Time of the Doctor"). Producer Philip Hinchcliffe has said, "We tried to get famous actors for the faces of the Doctor. But because no one would volunteer, we had to use backroom boys. And it is true to say that I attempted to imply that William Hartnell was not the first Doctor." The Doctor's previous incarnations are represented by images of production unit manager George Gallaccio, script editor Robert Holmes, production assistant Graeme Harper, director Douglas Camfield, producer Philip Hinchcliffe, production assistant Christopher Baker, writer Robert Banks Stewart and director Christopher Barry.
However, I do have a fan theory that the Doctor is more than a normal Gallifrean and more than a Gallifrean Time Lord. My theory is that the Doctor is actually a non Gallifrean who has an infinite number of lives, being reincarnated when he dies or is killed, unlike a Gallifrean who can not regenerate if killed. This hypothetical alien has been reincarnated as a Gallifrean at least once before he was reincarnated as the Doctor. Thus the eight unknown human looking faces seen in "The Brain of Morbius" might be eight previous faces of Morbius, eight previous faces of that hypothetical alien from his previous incarnation as a Gallifrean Time Lord who could regenerate 12 times, or some mixture of faces of Morbius and faces from the Doctor's previous life.
If that theory is true, even if the Doctor runs out of Gallifrean regenerations and dies as a Gallifrean, he/she will be reincarnated and live again as either a Gallifrean or a member of some other species, since I don't know whether this hypothetical alien incarnates only as Gallifreans or also as members of other intelligent species.
There were a number of hints dropped in the era of the Seventh Doctor that he was more than "merely" a Time Lord.
Feeling that too much of the Doctor's backstory had been revealed by the Seventh Doctor's era, writers Andrew Cartmel, Ben Aaronovitch and Marc Platt developed a new direction for the series. Cartmel wished to restore the character's "awe, mystery and strength" and make him "once again more than a mere chump of a Time Lord" – an idea the media dubbed the "Cartmel Masterplan". Under Cartmel, the show foreshadowed this concept; however, its 1989 cancellation meant that it was never realised onscreen. The proposed backstory was fully explored in Platt's 1997 novel Lungbarrow, where the Doctor is revealed as "the Other", a mysterious figure in Gallifreyan lore who co-founded Time Lord society with Rassilon and Omega. After a curse renders Gallifrey sterile, the Other devises biotechnological looms to "weave" new Time Lords; his granddaughter Susan is Gallifrey's last natural child. To escape a civil war with Rassilon, the Other throws himself into the loom system, where he is disintegrated and later woven into the Doctor.
This backstory is partially, but not entirely, consistent with my hypothetical alien who reincarnated as two or more Gallifreans with 13 bodies each. But it was never made explicit in television canon, though a few vague hints in the 7th Doctor Episodes are canon and much more consistent with my hypothesis which isn't very specific about the Doctor's previous lives before the William Hartnell Doctor.
One problem with the Doctor's biography is the constant time travel in the series, and suspicions that some episodes might be ind different alternate universes than other episodes, and the possibility that history has often been changed, including facts about the Doctor's biography. Nobody can be certain that facts about the Doctor's biography in one episode are canonical and true in other episodes.
Warning! Spoilers follow!
In the 18th season, in "Logopolis", written by Christopher H. Bidmead, 28 February to 21 March 1981, the Fourth Doctor occasionally spotted a mysterious figure, "The Watcher". The Doctor is fatally injured and "The Watcher" appears and merges with him, and then he starts to regenerate into the Fifth Doctor (Peter Davidson). "The Watcher" is considered m a mysterious"in-between incarnation" of the Doctor. So the Doctor was apparently in two places at once during parts of "Logopolis".
In season 23 of the Classic Who, with the overall title of "Trial of a Time Lord", the Doctor was put on trail for various alleged crimes by the Time Lords. The time Lord prosecutor was the Valyard.
The Valeyard (/ˈvæliɑːd/) is a fictional character from the long-running British science fiction television series, Doctor Who. He is described by the Master as an amalgamation of the Doctor's darker sides from between his twelfth and final incarnations. In the serial The Trial of a Time Lord, comprising the whole of Season 23 of the series, the High Council of the Time Lords appoint the Valeyard as prosecutor at the Sixth Doctor's trial, hoping to have him executed and thereby remove the sole witness to their near destruction of life on Earth.
At the time of "Trial of A Time Lord" fans assumed that if the Doctor would have only 13 bodies the Valeyard would be an intermediate step between body number 12 and body number 13. Body 12.5, I guess. But possibly some fans suspected that the Doctor might somehow get a new set of regenerations, as in fact happened decades later.
As of now, the Doctor has a new set of 12 regenerations and 13 bodies, and the Doctor's "last regeneration" would be his 26th body. So the Valeyard would be the combination of all the dark personality traits of every Doctor between the 12th or 13th body an the 26th body. If the Master was telling the truth, that is.
So is the Valeyard a past or future life, or both, for the present Doctor? Or is the Valeyard non existent in the fictional universe the present Doctor is in?
When the new Doctor Who began in 2005, everyone assumed that the Christopher Eccleston Doctor was the ninth one, after eight regenerations. But it was later revealed that the Eighth Doctor (Paul McGann) had regenerated into the War Doctor (John Hurt), who was thus the real Ninth Doctor, and that the War Doctor had regenerated into the Christopher Eccleston Doctor who was thus actually the Tenth Doctor. So if you see a reference to the Ninth Doctor, or a later once, you need to be certain which Doctor is counted as the Ninth Doctor in that source.
So there is same evidence that besides having had one set of 13 bodies as a Gallifrean Time Lord, and starting on a second set of 12 regenrations and 13 bodies as a Gallifrean, the Doctor might have had a number of previous lives before the First Doctor, the William Hartnell One. And there have been examples of "in-between" Doctors, alleged Doctors allegedly amalgamated from the personality traits of several future Doctors, possible alternate universe Doctors, and previous Doctors whose existence is only revealed to the audience later.
So good luck figuring out whether the Doctor had a previous incarnation cycle before the William Hartnell Doctor, as you ask and I have theorized for different reasons, or if Jo Martin's Doctor is one of those other possibilities previously seen, or possibly something else, until it is revealed in some future episode. There are too many possibilities in the Doctor Who universe for you or me to deduce what brilliant or silly plot the writers have planned.