I used to hunt that exact country. I lived in Carbondale area for 35 years. I have called all over the Lincoln Creek, Richmond Ridge, Lost Man, Independence, The Crown, Missouri heights, Reudi, Four Mile, Basalt Mountain, Avalanche Creek, and Cottonwood pass. Coyote hunting got really popular several years ago and all those places got much harder to hunt, lots of newbies standing out in the open with a caller in their hand, or calling from the seat of the truck. Everybody and their brother seemed to be a coyote hunter around 2019 to 2023. I killed lots of coyotes in some of those areas, enough that I think I damaged the population, but it probably recovers quickly.
I never did any good while using a snowmobile to access the high country, day or night, NEVER. Even hiking a mile or more away from the snowmobile. A lot of the truly high country, like above 9000' in the winter does not hold many animals of any kind. The snow is too deep for coyotes or bobcats to get down to mice, voles, and squirrel like rodents. In the winter I hunted on foot (snowshoes) from about 8500' and lower. Most of the coyotes I killed, averaged about 30 per year, were in the 7000 to 8000' range.
I hunted the high country in the early fall, sometimes with great success. I've killed them at 12,500' in September and October. But, there aren't that many coyotes living up there even in the summer. I would go to an area the first year and kill 3 or 4, the next year maybe 1 or 2, the 3rd year was usually not worth going.
Might be fun to try hunting at 9500' and higher in the winter, but far more effort for far fewer coyotes. You might see lots of coyote tracks on the snowmobile tracks, that's because snowmobile tracks are highways for them. And one coyote can make a lot of tracks.