We live 90 miles from the west entrance of Yellowstone, and been there many times. We were there last September, weather was nice, but below or close to freezing at night. The weather changes fast here in eastern Idaho, and faster in the park. September would be a roll-of-the-dice as far as weather goes.
We like to go just after the park opens very few people, mostly locals, and see the orange colored buffalo caves. Seems to be more animals moving around also. Not like the old days when the bears would give your car a big hug, have not seen a bear on the road in years. Have seen two people rubber necking, drive off the road into the river, one on a motorcycle with a kid on the back, the other was a van. There was a Ranger in front of us when the motorcycle went off, the river was only about four foot deep. He probably got a ticket for polluting, they get very unhappy about that sort of thing.
One other tip: Don't go buzzing around a blind bend in the road, take it slow. There is a good chance some guy was not thinking and stopped in the middle of the road to look at an animal, and you'll suck-up the seat cover when you find him, or worse.
1998 Four Winds 5000 Rigby, ID