Let’s talk about apocalypses and their aftermath in one place, because the borderline is the perfect example of “fuzzy.”
Who decides when characters are in an apocalypse versus when the apocalypse is over? If the apocalypse is the alien invasion, and the aliens win and take over the world, when are we “post-apocalyptic”? If the zombies are not eradicated, but only an occasional coastal problem, when is the zombie apocalypse over — or when is this the new normal?
If everything to speak of, starting with all intelligent life, is destroyed in the apocalypse, this gets easy: there is no post-apocalypse.
On the other hand, a huge amount of dystopic future fiction mutters about some vague apocalypse or breakdown or something that destroyed the global society we know, making space for unpleasantness in the future. If we make everything set in a dystopic future “post-apocalypse fiction,” then it rather loses meaning.
Apocalyptic stories have been a steady part of science fiction for nearly two centuries — about as long as there’s been science fiction under other names.
This is the basic good time of destroying your world for fun and profit, but mainly fun. H. G. Wells said he had a grand time bicycling his neighborhood and deciding which building would be destroyed by the Martians in War of the Worlds. Nowadays zombie apocalypses are all over the place (but World War Z is extremely good, even if zombies normally make your eyes roll), but there’s still room to battle the fall of civilization through plague, comets, rising sea level, desertification, human infertility (The Children of Men or Feed by M. T. Anderson), and a host of other classic reasons.
Closely related to these are the Last Man stories, where only one person is left alive (The Last Man by Mary Shelley, oddly enough, but also The Purple Cloud by Shiels) or where the Earth is dying, though humanity may still have centuries to scrabble through (Le dernier homme, or The Dark Land, or The Dying Earth by Vance or Zothique by Clark Ashton Smith).
There’s also the apocalypse that’s an Imaginary War. Not just any normal, ordinary war with conventional weapons, like The Battle of Dorking in its day, but one with speculative weapons. Aliens invade! (Yes, War of the Worlds and all of its near kin like the movie Independence Day, but nothing said an apocalyptic story can’t be fantasy, with a breakdown of barriers keeping out the demons.) New technology transforms warfare! And maybe magic is the new technology.
So you already destroyed the world and some generations ago at that. Now your characters survive and investigate the changed world, and scavenge in the ruins of New York or London. After London may have been the first of these, its less than hyper-competent hero trying to stay alive in foreign kingdoms in what was once England.
The world doesn’t have to revert to the Stone Age or the Middle Ages. Earth recovering from some breakdown, and it may have been social/political, is the setting for The Handmaid’s Tale or The Hunger Games, Disruptor or Star Man’s Son, The Book of Rack the Healer or A Canticle for Leibowitz. An alien invasion long ago sets up Omha Abides or other stories.
Like an alien world settled by humans, but with less hand-waving about lost colonies and forgotten FTL technology, post-apocalyptic settings let you have our unfortunate descendants stopping disasters and reversing dystopias in a society not ours, but rooted in our familiar world, or dealing with a changed ecology that can become the main villain.
Engine Summer by John Crowley shows different groups living in the remains of our infrastructure, but not really remembering much about us.