I can think of four specific and very different "post-apocalypse" (PA) settings that show differences in how one could approach the genre.
The Morrow Project is a straight-up PA setting. Except the PCs aren't from that setting. They were put in stasis before the fall. The idea is they are a well-trained, well equipped team that's supposed to help rebuild things immediately after the collapse. Only… they don't wake up for 150 years. This is closer to what you are saying you don't want. This is kind of what I consider a "baseline" for PA.
Deadlands is a setting that sort of resembled wild west fiction, except with magic. Technically, it isn't considered a (PA) setting. But if you look at the game's timeline, magic is introduced. Chaos ensued. Now you have magic-users and gunslingers and shaman and… Yeah. It is a PA world. Or it would be to anyone who lived through the introduction of magic. The idea here would be to explore not the PA aspects, but how life differs in the 1800s of this world with the 1800s of real history.
ShadowRun like Deadlands, introduces magic to a futuristic sci-fi world in a big cataclysmic event. Again not technically a PA RPG, but given how dark the world is in SR, I think the label fits. By shifting from the out-of-box timeline to one right after the introduction of magic into the world, it would be hard to argue against it as a PA setting. I mean, nations collapsed, entire regions of the world became uninhabitable… So again we could use this as a way to explore how the mega-corps rose to power over nations, and how the fantasy elements vied for power with existing entities.
Another take is Numenera. This is closer to a true fantasy world, except that artifacts are everywhere from a previous, high-tech, age. That age left behind things like nanobots and other technobabbly-goodness that now acts like magic, since no one can really understand it anymore. Again, though, it isn't technically presented as a PA setting. It was more of a "high fantasy RPGs are quirky because they introduce so many historic anachronisms. So solved that by…" But given that the previous ages were high-science and this age is basically not, it can be considered in the PA realm.
I think the above advice, "Don't say apocalypse", is key, no matter what. Sure, you need to keep at least some tropes, like the difficulties in acquiring high-end goods without a global economy, or the lack of high quality medicines, or the reduction in personal safety… But if you just set the game in a smallish town, with no mention of an apocalypse, then the players will explore that world without acting out the tropes. Eventually, they'll figure it out if they explore well enough and you describe it right. But they won't know at first.
Another idea that may help you is to look at the key, obvious, tropes from the PA worlds, then decide how to subvert those tropes. Especially the ones that don't make sense to you. Like "Mad Max" has muscle cars that get like 5 miles to the gallon, and everyone has one and everyone guns their engines constantly like gas is essentially free. So how does that work? What's the trick here? How can you use that to mess with the heads of anyone trying to understand the PA genre you're using?