Slingshot Dakota started off as a three piece emo band from Brooklyn, NY/Easton, PA, but after the departure of their guitarist they continued on as a duo. Piano, and a drum. A fucking ballsy piano mind you, and a goddamn amazing drummer. This is nothing like The Dresden Dolls, this is a completely different story. Their sound is a combination of punk and indie pop. Even if this doesn't sound appealing, I suggest you at least give it a shot.
Prawn is a fucking emo/post-rock band. They're goddamn amazing, and if you don't like them, I want you to die.
The Burning Paris released two albums, and then an anthology, and presumably called it quits after. The fact that they are not world renown is astonishing to me, because this is one of the greatest bands I have ever heard in my entire fucking life.. and I just found out about them yesterday.
1. I Drive
The Little Explorer was a screamo/emo (with other elements) band from the United Kingdom. They were around from 2002 to 2008, and released one album, and a demo.. thing. Extremely talented, and their replay value is relatively high. I've searched a while for both of these releases, and they're hard to come by. It's a good thing this band is amazing, or I wouldn't even bother putting these up.
I Create is from Woodstock, Georgia. They're a post-rock screamo band, and if you don't like them you can just go fuck yourself. This is one of the best, one of the greatest. They're seriously too good to be this unknown. They're talking about breaking up, and I'm gonna fucking punch them in the face.
Hour of the Wolf is a hardcore band from Prescott, Arizona. They're slowly forming into my favorite hardcore band, so if you want to prevent that.. top this band. They've been compared to Black Flag, but fuck that, no. 
Burn Her Letters is a screamo band from Eskiseher, Turkey. They formed in 2006, and share their music freely. That's right, they're one of those awesome bands who exist for music & music alone. Their MySpace is not in English, so.. I can't tell you anything about them. I've only heard one song, but I got all their material, because I really loved that one song; I tend to do that a lot. I'm sharing the experience with you! :D 
Marquette, uh.. I know nothing of them, and I haven't even listened to their two releases yet. So, why am I recommending them? I was extremely impressed with one of their songs that I'm 90% sure all of this will be amazing. Emo, yeah, jazz..?.. was mentioned, but.. I doubt it. I'm pretty sure they're just.. emo.. and maybe some offshoots of something else, but I highly doubt jazz.
Snowing is one of the best emo bands to ever come around, and just.. so far, one of the best bands in general. Some people are familiar with their work, but it's my opinion that not enough are. From Leigh Valley, Pennsylvania, these boys know how to make memorable music. Everything about them is perfect, and I honestly have no complaint about them. Not even something they could improve on; they're that good. All of this opinion, of course, but fuuuuckkkkk-- you have to like these guys.
The Tiger Lillies have been around since 1989. They're based in England, and are known for their decadent live shows, and festive music. They fuse dark cabaret, folk, and circus punk. I've known about these guys for about 5 years now, but I've recently acquired a decent collection of their music. They have 25 major releases, so.. it's quite an extensive library. 