As with any tour, especially those involving elevations, the keyword for packing is -- layers.
At the beginning of October you can expect some rather cool temperatures at the elevations you will be traveling. You can expect fresh snow in the Big Horn Mountains, Yellowstone Park, and, perhaps, even Bryce Canyon. Oh, also Park City, Utah. But you can choose how much interaction you have with the snow and the cold.
Conversely, at Lake Powell it will probably be in the high 80s or maybe even 90 degrees. so you will want shorts, tee shirts, and sandals for those two days.
The tour is bus-intensive (some 2,500 miles) but the pacing is good. The only objectionable bus day I recall was from Lake Powell to Durango, but you will have spent two days in Lake Powell and will spend two days in Durango. The lunch break at Guldings in the Monument Valley is a refreshing break in the middle of the long trek.
When we took GCT's National Parks tour we knew what to expect. It was a repeat, in reverse, of a trip we had taken nine years before. But it was at a different time of year and that made a lot of deifference.
The Program Director and coach driver will make or break the trip. But from my experience with GCT (ten trips) I feel certain that they will be quite good. If anything at all is bothering you during the trip, give the professionals an opportunity to rectify the situation.