Dobie Theater is a good place to catch independent films year-round. It's not the most comfortable or nicest-looking theater, but you'll definitely get your money's worth when you can't find that movie playing anywhere else.
They also have special events showing old movies and their midnight shows are also kind of fun. If you're a UT student, then this theater is convenient since you don't have to drive anywhere. If you're not, though, then you can park on the garage across the street. Just make sure to get your parking validated when you buy your tickets.