Although very beautiful, the Lewis River is out of the way for most fly fishers. Unless you are traveling to or from Jackson and the Teton area it may not be worth your time. I have fished both the Lewis River and, in the fall, the stretch between Shoshone Lake and Lewis Lake called "the Lewis channel". The Channel is best accessed by boat via Lewis Lake and in the fall many large browns migrate up the channel from Lewis Lake to spawn.
I have, on occasion, found good size brown trout rising to dry flies. It may take some walking and looking. There are green drakes in late July.
Lewis River - by Montana Angler
Fly Fishing The Lewis River - by Fly Fishing Yellowstone National Park