Robert Heinlein's The Door Into Summer
At the time of the life of the protagonist when the story starts, there is no time-travel. But there is already cryogenics : one can get "frozen" to "sleep" for decades and be "thawed" no older than one started.
For some reason I forgot the protagonist decides to "sleep frozen" for a few decades.
The novel opens in 1970 with Daniel Boone Davis, an engineer and
inventor, well into a long drinking binge. He has lost his company,
Hired Girl, Inc., to his partner Miles Gentry and the company
bookkeeper, Belle Darkin. She had been Dan's fiancée, deceiving him
into giving her enough voting stock to allow her and Miles to seize
control. Dan's only friend in the world is his cat, "Pete" (short for
Petronius the Arbiter), a feisty tomcat who hates going outdoors in
the snow.
Hired Girl, Inc. manufactures robot vacuum cleaners, but Dan had been
developing a new line of all-purpose household robots, Flexible Frank,
when Miles announces his intention to sell the company (and Flexible
Frank) to Mannix Enterprises in which Miles would become a vice
president. Wishing to stay independent, Dan opposes the takeover, but
is outvoted and then fired as Chief Engineer. Left with a large
financial settlement, and his remaining Hired Girl stock, he elects to
take "cold sleep" (suspended animation), hoping to wake up thirty
years later to a brighter future. The examining doctor at the cold
sleep facility immediately sees that Dan has been drinking. He warns
him to show up sober or not at all 24 hours later for the actual
procedure.
When he is "thawed", he finds out that things are not, for him, the way he expected
Dan wakes up in the year 2000 with no money to his name and no idea
how to find the people he once knew. What little money Belle let him
keep went with the collapse of Mannix in 1987. He has lost Pete the
cat, who fled Miles' house after Dan was drugged, and has no idea how
to find a now middle-aged Ricky.
As it happens, time-travel now exists, but it is experimental and IIRC illegal. Still, he manages to travel back to his original time. There he does something that in fact had already happened in the future he has lived in after his first "thawing" (there is no time-paradox).
In Boulder, he befriends Dr. Twitchell, a once-brilliant scientist
reduced to drinking away his frustrations. Eventually, Twitchell
admits to having created a time machine of sorts. With the machine
powered up, Dan goads Twitchell into sending him back to 1970, some
months before his confrontation with Miles and Belle. He materializes
in a Denver naturist retreat in front of a couple, John and Jenny
Sutton, whom he befriends.
Having done it, since time-travel does not exist in his own time, he has to get "frozen" again but when he is "thawed" for the second time, things are OK for him.
With Pete in his arms, he sleeps for the second time until 2001. He
greets Ricky, now twenty-one, when she awakes. They leave for Brawley
to retrieve her possessions from storage, and then are married in
Yuma. Setting himself up as an independent inventor, he uses Ricky's
Hired Girl stock to make changes at Geary, settling back to watch the
healthy competition with Aladdin.
As far as being a Great Classic Novel, Door into Summer was voted as one of the top 50 SF novels of all times in three different Locus surveys