We were a little nervous from some of the reviews and unsure about going to the area at that time of year, but decided to book anyway with the Rustic Inn for a quick fall trip to Jackson Hole in early October 2012. We found a great rate online and were pleased to be upgraded upon our arrival. The grounds are beautiful, the rooms cute and well kept, and the location great within walking distance to downtown Jackson, across the street from the Visitor's Center, and the same road leading to the parks and out of town. The breakfast was very handy and quite good. We had a great time overall and would highly recommend the Rustic Inn for couples or families. Our only regret was missing the fall leaf changes (a little too late) and not seeing a moose!
As far as visiting Jackson Hole, Grand Teton, and Yellowstone in early October, we had a great time. There isn't as much open (fewer activities like rafting or the playhouse), we were a little too early for some of the animal migrations, but we still had plenty to do, the weather was nice (we hiked without jackets), and we even got some great deals in town (like buy one get one free meals at one local steakhouse!). I think you'll like this area any time of year!