Bechler River

I guess you could call the Bechler  River a "hidden gem" in many way. It is a very good trout stream holding large rainbows and even bigger cutthroat.  It is probably a two hour drive from West Yellowstone. If you make the drive, you will not have much competition from other fly fishermen. After the drive, there is a 3-4 mile hike from the ranger station to "the meadow section" along the Boundary Creek trail. From my somewhat limited experiences, the meadow section holds the bigger fish.

I should also say that these fish can be very picky but you may very well be rewarded with a 20" rainbow or cutthroat for your efforts. Again, my success was primarily with terrestrial patterns,

If you go to this area, you will also have decent access to Fall River or Falls River along your route. Although the fish tend to be fairly small, they will readily eat a fly. 

To be perfectly honest, if you are driving from the North (West Yellowstone, Mt) you will have to drive right past the very best (my opinion) trout stream in the US...the Henry's Fork. If you have not fished the Henry's Fork and just drive past it, you are a fool!

More information:

Fly Fishing The Bechler River - Fly Fishing Yellowstone National Park

Bechler River - Montana Angler

DIY Guide to Fly Fishing Bechler River in Yellowstone National Park - by DIYflyfishing.com