Lots of places you could go, for sure - I'm just going to discuss the Sierra as that's where I have lots of experience.
In the Sierra, Mid-September thru Mid-october is a great time to be out and a chancy time to be out. Great because the weather is often very nice - comfortable days, cool evenings and frosty mornings - and there are very few people out and about. Chancy because a storm can move in and dump rain or snow on you in large quantities.
Were I planning a Sierra trip for that time frame, I would not plan a mileage-making trip, nor a trip that took me deep into the mountains. I'd go somewhere where it's easy to bail if a storm dumps 2 feet of snow on you - which can and does happen that time of year. And I'd watch the forecast closely before the trip. If you have the schedule freedom to plan a trip for that one-month window, and then take the actual trip in the week that has the good forecast, then I can think of lots of places to go. But if you have to pick a week now, and take the weather as it comes, then I'd suggest having a few options laid out so you can adapt to the weather.