The Madison in the park should have some lake run fish by September, and those are typically big fish. Browns spawn, bows give chase to eat the eggs (hell of a way to make a living.) Dry fly fishing on the Firehole could be starting then as well, mostly for modest sized fish. Otherwise best Sept options in the park are the NE options McNerney mentions. (Keep in mind that all of these locations are very well known, and you will sometimes have lotsa company fishing.)
West of West Yellowstone, the Madison can come to life in the fall as well. Your timing could be a perfect overlap of summer terrestrials and fall fishing, OR, a perfect gap between the two. But worth keeping an eye on it.
Blue Ribbon Flies is a good source of info for west side YNP and Madison info, as well as NE YNP. Galloups Slide Inn is on the Madison and they are knowledgeable about the Madison.