I second OC's recommendation of West Yellowstone as a place to base your trip around for all the reasons he stated. In addition to Blue Ribbon fly Shop, another great shop in West Yellowstone is Bob Jacklin's. Jacklin, like Craig Matthews the owner of Blue Ribbon, is a long-time, year-round resident of West Yellowstone and both of them have fished the waters in the area for many, many years.
Missoula would be my second choice of place to base your trip from because there are many rivers and lakes within 1.5 hours of Missoula. There are the Clark Fork River (above town all the way to Warm Springs some 70 miles distant upriver, in town, and below town all the way to St. Regis some 70 miles downriver), the St. Regis River, the Bitteroot River, The Blackfoot River, the Jocko River (an overlooked superb trout fishery 30 miles north of Missoula on the Flathead Indian Reservation), Rock Creek, the Little Blackfoot River, Flint Creek, Montana's Clearwater River, Swan River, Lolo Creek, and many other smaller, little known streams and all of these rivers and creeks have excellent blue wing olive hatches and tan caddis egg laying flights along with Tricos in September. There are also many fine lakes within 1.5 hours of Missoula beginning with Mission Lake on the Flathead Reservation, Swan Lake, and the huge Flathead Lake itself.
The Bighorn is very nice in September because the summer crowds have left; but it is a long way from Billings (90+ miles depending on where you fish it) and it has limited access unless you hire a guide and float it. Billings is not a good place to base your trip from because the closest best river is the Stillwater (45-65 miles depending on what section you fish) and it has limited public access. And the Yellowstone in the Billings area is marginal trout water, there are trout there, but it is a very large river in Billings and the trout are small in number there and very spread out.
The Helena area has more than just the Missouri, which 25 years ago was a tremendous fall migrating brown trout fishery in the 3 miles of river between Hauser and Holter Dams, but this fishery was ruined with the inadvertant introduction of Kokanee Salmon to Hauser Dam and Holter Dam when an irrigation reservoir was drained. The Little Blackfoot, upper big Blackfoot, Little Prickly Pear Creek, and Prickly Pear Creek are all within 40 miles of Helena. If you travel 1.5 hours, the lower Gallatin, Jefferson, lower Madison, and upper Clark Fork are within reach.