Personally, I'd suggest one of the late hunts. December or January. The elk will be in the low sage country. Some roads may be snowed in, but you should be able to drive to spots to glass from, then hike in once you get a herd spotted. Camping wouldn't be much fun, but hotels in town are available. If you're intent on hunting the high mountains, then this probably won't be your best option, though you'll see the mountains for sure.
The extra cow hunts during October can be nice, but you'll likely have lots of competition from the general season hunters. Some areas such as 94 you can avoid this though as the general season for bulls is only 10 or so days long and that's when the vast majority of people will be there.
