L. Sprague de Camp and Lin Carter took a stab (pun intended) at a story set in the “New World” during the Hyborian Age:
It’s been decades since I read this novel so I can’t speak to its quality. It’s not REH, of course, so there’s that too. Judging by the names in the Wikipedia synopsis, it seems to have a Meso-American vibe. If you’re looking for something more in tune with native North American cultures, it may not prove very inspirational.