I used to go to the winds every year, preferably in September. There is less bugs (hopefully), cool days and possible some fall colors and or early snow. The last time I was at the winds, it was the 3rd week in October and we had traces of snow and pretty cool nights.
The only draw back is elk hunting season when every hunter in Wyoming seems to be up there. I don't recall when the season starts/ends, though. Still, it is pretty easy to find plenty of isolated, grand alpine peaks and valleys. If it were me, I'd head straight into the Cirque of the Towers, approaching from the west side out of Pinedale, WY.
Have a great trip. I'm jealous!