S Central Idaho, where I hunt, the first good snow, even 3"-4", will start them moving, anytime after the first week of Oct. Without ongoing storms they will take their time, week-ten days to get near wintering grounds. If weather holds till last week of oct, they start anyway, but can be down in a day to day and a half. That's 50 mi as crow flies over steeeep mountains.
We hear lot's of stories about guys hunting all the way till thanksgiving up high, but not any more. Something changed.
All that said, those big boys still seem to hold back or at least stay hidden until the rut in mid-Nov.
I agree with BPK and Heartshot. I took my best General season buck on the 12th of October, and he was already migrating with other bucks. In most years, they seem to migrate near the beginning of October, but will hang in the timber at mid elevations.