Hey, popmuze--I'm not sure I understand what you're trying to market--non-genre, non-mainstream, sort of quirky, semi-literary? Sheesh!--but I do know something about some of these presses, so here's a few pointers:
Graywolf: ultra-literary and very selective. Not many titles per year. Great rep, but generally not what you'd call 'quirky.' Don't need an agent, but a pile of publications in literary mags would help. Semi-literary probably won't fly.
Soft Skull: really edgy. Small. Weird and cool. Don't need agent.
Akashic: edgy, plus mystery. Some folks love them, some don't care for them, but in any case they have been closed to subs for quite a while, agent or no.
MacAdam/Cage: A really big small press with great distribution and massive support of their backlist. Literary, but including quirky, offbeat, and mystery. Check out "Time Traveller's Wife" or "The Rabbit Factory." These guys are gradually turning into an alternative to the 212 area-code conglomerates, and may save thereby Western Civilization. Don't have to have an agent but it sure helps--they have interns read their slush piles, and some of said interns are idiots. (Including the infamous Logan Rapp.)
Soho: Great small press. Literary and mysteries (though they claim their mysteries are all literary. Uh-huh.) Don't need an agent.
Mid-list: hard to get published there unless you've already been published elsewhere.
As far as I know these folks all pay advances, but I've never seen their contracts. Most small presses pay advances, but they are often nominal.
Good luck--and I look forward to reading your book just to find out what non-genre non-mainstream is!