Weather in Jackson, WY
Weather in October in Jackson, WY averages about the same as Austin in January, so plan accordingly!
Currently, the forecast looks promising with highs in the mid-60s and lows in the lower 30s. Still, remember that a low humidity, high wind 60-degree day can feel frigid to us Texas folk. As of this writing, no rain is expected!
As a long-time backpacker and former resident of Colorado and Maine, my suggestion is always layers! A big parka will be overkill in all situations except maybe sitting outside at night waiting to see if we can spot the Aurora Borealis. A long-sleeved non-cotton base layer with a hoodie or fleece sweater over it will probably be fine in most cases. Just adding a windbreaker/raincoat over that will get most people down to 20-30 degrees if they're active.
The Weather Channel
https://www.accuweather.com/en/us/jackson/83001/daily-weather-forecast/336564