What to do with a stray cat? What are the real options?

EDIT! Halo came through! I took the kitty to their facility Sunday afternoon. Thanks for all the help everyone, all of metro Phoenix's stray creatures are better off with all of you looking out for them. If anyone is interested, Halo will post him on their site in a day or two, and his new name will be Timone. A couple months ago a stray neighborhood cat yelled at our door (for well over an hour) until we finally let him in for a minute. "A minute," right. I have no idea why he picked our place, but I assume he must have smelled the three cats that live here that actually belong to us? Anyway ... what's the best way to find this sweet guy a decent home? Now, "post online to one of the lost pet sites!!" is gonna be everyone's favorite response, but it shouldn't be, because 1) I've tried that, of course, and 2) it does nothing. Well, most often nothing. I've tried the "lovelost" site, and PetCo's site (which actually IS the lovelost site, and that prompted me to create duplicate accounts), and even the local NextDoor app. And Facebook. All of them suggest posting to each other, and all of them are already absolutely full of dozens of found / stray pets that look almost identical. There might be two or three people that have lost a pet and are actually taking steps to search for them, but the latter is vastly outnumbered by the former. I love that there are occasional success stories of reunited pets through those sites, but those, unfortunately, are outlier situations. Sad fact: there's a huge, huge population of stray cats that nobody is looking for. I could just drop him off at a shelter. My cats do not like him, and have started peeing in places they shouldn't in protest (or confusion; I don't have e