Gareth Rees has provided a term which may be used (albeit perhaps not in widepread use) for the specific subgenre you're asking about. But in general, stories of this type are still called post-apocalyptic fiction.
From the Wikipedia page on Apocalyptic and post-apocalyptic fiction:
The story may involve attempts to prevent an apocalypse event, deal with the impact and consequences of the event itself, or it may be post-apocalyptic, set after the event. The time frame may be immediately after the catastrophe, focusing on the travails or psychology of survivors, the way to maintain the human race alive and together as one, or considerably later, often including the theme that the existence of pre-catastrophe civilization has been forgotten (or mythologized). Post-apocalyptic stories often take place in a non-technological future world, or a world where only scattered elements of society and technology remain.
Here it seems clear that "apocalyptic" would be for stories that take place during, or perhaps in the immediate aftermath of, an apocalypse, while "post-apocalyptic" is still used for stories set generations or eons after the apocalypse.
Other sources agree:
Post-apocalyptic fiction can be set in the current day or the far off future. Additionally, the story can take place right after the cataclysmic event or years after the event. In post-apocalyptic novels, technology can be that which we have never seen before, or there can be no technology at all. Also, characters can remember what the world was like, or they can’t remember at all what the world was like and will fantasize about the way it used to be or even go so far as to create myths about the world before the destruction (often our current day).
-- Young Adult Library Services Association
Myth Builders' article abdout post-apocalyptic fiction mentions several stories, such as A Canticle for Leibowitz, which, despite being set centuries after the apocalypse, are still widely accepted as works of post-apocalyptic literature.
TL;DR: regardless of whether or not this subgenre can be given its own name, it does count as part of post-apocalyptic fiction, not a separate thing.