Its a crap shoot. Like I mentioned, last year even in late Sept it snowed a bunch at higher altitudes and on my last day in Glacier, the Going to The Sun Road was closed at the Avalanche Lake trailhead, which means you arent seeing a good chunk of the park. It opened up again several days later, but it would suck to plan a trip and get snowed out. By this time of year, most of the campgrounds are closed because they know that the potential is there for some snowdumps. But the 10 day forecast looks pretty good this year into early Oct.
Also some years, they close that road early due to road construction/maintenance they try to squeeze in before the snows come. If that's the case, access from one side to Logan's pass will be fine, while access from the other side is limited. (https://www.nps.gov/glac/learn/news/logan-pass-opening-and-closing-dates.htm).