In Philosopher’s Stone, when Hagrid introduces Quirrell, he says:
“…He was fine while he was studyin’ outta books but then took a year off ter get some first-hand experience…”
If Quirrell took a year off, this means that he was at Hogwarts more than a year ago. Voldemort cursed the DADA post when Dumbledore denied him as a teacher, so that nobody could stay in the position for more than a year.
So who was the DADA teacher in the year that Quirrell was gone?
J.K. Rowling clarified in an interview that Quirrell did not previously hold the Defense Against the Dark Arts position before Harry's first year:
Incidentally, I know some have asked about Quirrell with regard to this question [about the Defense Against the Dark Arts position being cursed]. He was teaching at Hogwarts for more than a year, but NOT in the post of D.A.D.A. teacher. He was previously Muggle Studies professor.
As for who was in the position before Quirrell, the short answer is that the individual has never been identified. There have only been nine DADA professors identified by name. Six were present during Harry's six years at Hogwarts (Quirrell, Lockhart, Lupin, Crouch, Umbridge, Snape), Alastor Moody accepted the position before being impersonated by Bartemius Crouch Jr., Amycus Carrow (a Death Eater) was installed during the time of The Deathly Hallows, and Galatea Merrythought was the teacher who retired while Tom Riddle was in school.
While J.K. Rowling alluded to a multi-year professor after the events of the story (thanks to the curse being lifted by the defeat of Voldemort), she never gave him or her a name. On the other end of the timeline, Rowling has never discussed, much less named, the professor before Quirrell.
So unless Rowling changes her mind and gives the individual a name, we will never know who the Defense Against the Dark Arts professor was before Quirrell. I suspect that it's not something she has ever considered herself, as the person served before Harry arrived at Hogwarts and she never intended that professor to have a legacy that would impact Harry or the others at Hogwarts.
The way that I understood what Hagrid said is that Quirrel was studying books on his own. It would seem in the Wizarding world that because there is such a small population of magical folk that everyone knows everyone else and their business, much like what you see in a small town. I do not believe that Quirrel held a position at Hogwarts but studied magical creatures in England, then took a year to travel and discover these creatures on his own.
I know that Quirrel was NOT the D.A.D.A. teacher before, because it was proven that he was the muggle studies teacher. So perhaps it was still like how Harry experienced Hogwarts, with a new teacher every year. Although many beliefs are probable, Dumbledore said that there was a curse on the position, so it is likely that there was a new professor every semester.
The individual holding that post in the year just before Harry's 1st ('90-'91) is not named yet. The only recent character, before Harry's time, to hold that post that we know of was Patricia Rakepick in '88-'89.
Patricia Rakepick appears as a major character in the game HP: Hogwarts Mystery. She attended Hogwarts School of Witchcraft and Wizardry from 1967 to 1974 & she was Sorted into Gryffindor. She arrived at Hogwarts as a "Curse-breaking consultant" in '87-'88 school year to consult on the Cursed Vaults at Hogwarts. She trained several students to help her deal with the Sleepwalking curse that had taken over Hogwarts that year. Later she took the position of DADA teacher in the following school year, '88-'89, mentored the group of students that broke the curse previous year in search for the next Cursed Vault. Due to events of the game, & the curse on the DADA position, did not continue after that school year.
