Check out mubi and filmstruck. Mubi has 30 movies at any given time. Every day the oldest drops off and a new one gets added. I've watched many movies there that otherwise would have been on my to watch list forever. I've discovered incredible, small movies because of it. Filmstruck has an enormous catalog of indie and old movies. Unfortunately both services happen to end up buffering much more frequently on my devices than the big streaming services do. Might be my connection though.
Look for indie movie theaters in your area. I'm a little spoiled in that the town I grew up in had an amazing indie film theater (Cinema Arts Centre) and now I live in NYC where I can go to the IFC Center or Sunshine Cinemas to see indie films.