The first one, Cosmic Incursion, adds a sixth player color, along with new aliens of course, and the Rewards deck, which is one of those things where once you play with it you don’t want to go back.
Other than stuff already mentioned (seconding Libertalia, Mysterium and Evolution in particular), my top pick would be Adrenaline. The expansion takes it up to 6 and gives you team play options.
The Great Split and The Great Fire of London 1666 are also good (though the latter has a fiddly set up).
And if you combine Bullet and Bullet you can play up to 8!
Lords of Vegas with the Up expansion can fit 6, and is likely really good fun (I have only gone to 5 so far, and it was a blast). Hard to get right now though if you don’t already have it.
Otherwise, I can only echo what others have said: Libertalia, Mysterium, Zoo Vadis, Heat. I suppose Flamme Rouge would count as well, with the expansion to add more cycling teams.
Shadows Over Camelot would also fit the bill, but again, hard to find now
Oh, Sheriff of Nottingham goes up to 6 with the expansion, at least in the first edition. I think the new edition includes some expansion content, but don’t know if it has the 6th player materials. May be worth looking at.
Ah, this is one that goes effortlessly to 6, and in fact the more crowded the space is the better, but doesn’t feel any more complicated. Definitely a good game at 6.
Other people have suggested Mysterium and Tsuro, and I agree. I’d also say my old favourite Coup; ideally Concept has a few more, but it can work all right at 6; Deep Sea Adventure, Freehand, though it’s more of a party game; Letter Jam; Pirate 21; Not Alone. (Some of these are fairly hard to find now, though they’re in my collection; I’m not saying you should go out and look for them.)
Food Chain Magnate goes to 6 with the expansion. Not played above 5 personally but can’t see a reason it wouldn’t work at 6 beyond some people won’t enjoy FCM.
Steampunk Rally either base or expandalone go to 6 and has the advantage of being a drafting game that’s largely simultaneous play so not much difference in downtime between 3 and 6 players. Best game I’ve had of it was at 6.
Cthulhu Wars is excellent at 6 but you need space and funds in serious quantity and therefore the enthusiasm for it. Base game is 4 player, you need the larger map and 2 faction expansions, so that’s outlay of around £200…
I find the down time between turns is usually really high in this game if there are too many players, but obviously YMMV. I don’t like to play with more than four.
If you play it with people who know the game, it generally flies by. BUT it does mean that you don’t actually get to put many tiles down at high player numbers so it can feel a bit pointless.