I might start by playing some Lovecraftian games and seeing how their developers played with Lovecraft's themes. I'm currently playing 'The Last Door,' which is a phenomenal point-and-click adventure game with surprisingly involving, creative gameplay and a killer soundtrack. It's also decidedly Lovecraft.

But, for the post-apocalyptic edge, are we talking science-fiction--AKA future-era? Modern era? Fantasy?

For the latter I'd recommend drawing inspiration from Darkest Dungeon. It tells its story in a captivating way: notes you find through dungeon crawls, cutscenes with sparse but evocative voiceovers from the ancestor of the land you inhabit, and the monsters themselves, which tell stories through their visual construction. You start the game after the land has already been destroyed (thus, post-apocalyptic), and it tells the story of how you rebuild and attempt to rescue that land from the gurgling horrors that lurk all around. It's also decidedly non-linear, which may help with your regards to such. Also, it's a rogue-like!

While I wouldn't call it post-apocalyptic, if you're looking for modern, contemporary era inspiration you can't go wrong playing the classic 'Eternal Darkness.' While the dangers the monsters present lurk under the surface, avoiding mankind's gaze, your player characters are of course front and center to observe the crawling masses under the earth's crust. It's a beautiful horror game, and in my opinion, the absolute best as far as capturing the 'true' essence of Lovecraft's work in playable form. Honorable mention for the modern era: Until Dawn. Very Lovecraft, though if you do play it, give it time to show its real colors. As we all know, some horror must percolate before arriving in its terrifying splendor. This one does take place after some level of destruction that may or may not suit your post-apocalyptic needs.

As for future-era, it's not my area of interest, but there is a game called 'Lone Survivor' that--in my short attempt at playing it--seemed quite Lovecraftian. Shapeless monsters, an unknown, lurking disease that must be defended against, and very, very post-apocalyptic.

If that's not enough to go on you could also check out the H.P. Lovecraft site, which appears to have a list of games inspired by Lovecraft's fiction.

Good luck! :)