In the mid to late 90s everyone did call June of 44 and such bands "post-rock".
Why do we need to invent new genres all the time. June of 44 and Mono can both be post-rock in the same way that the Beatles and the Stones can both be rock. Jeez. What's next, are we going to retroactively call Slint "post-metal"?
Actually, that's a genre already, if you can believe it.
Explosions in the Sky, while a pretty mainstream suggestion, is a pretty important one, too. Check them out.
Cue, Mogwai, If These Trees Could Talk, Red Sparowes, The American Dollar, Parabstruse, and Pg. Lost are all interesting. Give them a shot, and if you need help finding them, PM me.