Other than the Firehole River with its BWO and white miller activity ( don’t overlook offering soft hackled patterns!) making fish active, the fall season brown trout run is ongoing on all hosting streams. All other streams offer tough fishing because of cold, low water. If you seek brown trout, no matter where you choose to go: the Madison River (which also offers run-up rainbows), the Gibbon River, the Firehole River below the falls, the Lewis River (anywhere but the steep and dangerous canyon), or the Snake River at the South Entrance, brown trout are migrating. The only problem is that there are more fly-fishers than trout on all of these! Nevertheless streamer patterns should work well. But for the most heavily visited locations ( Lewis River Channel, just below Lewis Lake, the Madison River within the park, the Gardner River, the Gibbon River, fish have seen streamers presented by the mass of visitors and may associate them with vibrations and disturbances from wading visitors. So try smaller flies ( bead head nymphs, egg patterns, etc.). Old Lewis River favorites pretty much forgotten these days include the Montana Black Nymph and Bitch Creek Nymph. They sure worked well decades ago.
For all these streams, please avoid spawning locations: let’s do everything we can to help a successful spawning season, on which future populations depend. Remembering that the Yellowstone Park fishing season ends on the first Sunday in November and winter is just around the corner, you do not have much time left to encounter those big park browns.