Utah is the most stunning place in the US to hike, in my opinion. Mid-October is the perfect time to go there, as the terrible summer heat has gone.
I am not sure there are any long/unbroken 100-200 mile trails there, and even if there are, I'm not certain they'd traverse through the very best sections.
Get a permit (as soon as possible!) for a 4-5 trek through Grand Canyon National Park. Your itinerary would be: Fly into Las Vegas (cheap to fly there and cheap to rent a car); drive to Grand Canyon; hike GC. Then drive through Utah and hike there before looping back to Las Vegas. There are too many fantastic options to name in southern Utah: Grand Gulch, Paria/Buckskin Gulch, Escalante River, and several national parks including Canyonlands, Capitol Reef, Zion, and others, all perfect for backpacking in mid-October.
How much time do you have?