AWS and others already gave you great advice. I'll still put in my $0.02.
I grew up hunting all the areas around Rexburg and still hunt that area on a regular basis when I visit family. You have plenty of good places to call on public land. There are small hills, depressions and/or openings in almost all of that sage. There are many small hills in the sage country north of St. Anthony, around Mud Lake and Dubois that give you some elevation. As mentioned by others a shotgun can be helpful in the bigger sage because the coyotes could sneak into less than 20 yards before you see them. Those areas with big sage that are best hunted with a shotgun can be very productive because the coyotes feel safe coming in. In open areas they will hang up or do large circles to get the wind when coming in. There was an interesting study done on radio collared coyotes on the INEL West of Rexburg that showed how coyotes circle just out of sight.
There are coyotes all over the mountains in Idaho. However, I do think most of the best coyote habitat in Eastern Idaho is in the sage or has at least some sage near by. The Island Park area you mentioned does have a bunch of coyotes. There won't be as many coyotes in the thick timber areas but there will still be a few. If I was calling coyotes in Island Park I would be focusing on areas that have sage and clear cuts. With a predator call in Island Park you never know what might show up. You could have a coyote, fox, bobcat, wolf, black bear, grizzly bear or mountain lion come in to your call. Personally I wouldn't want to call in a grizzly bear to 15 yards when all I have is my 223. I would make some adjustments to the areas I choose to call in Island Park. In grizzly country I would also think about bringing a friend, a bigger gun and bear spray.
If you are looking for areas to go, send me a message and I could give you a number of ideas.
Attacked is a picture of an Eastern Idaho coyote that I came across while scouting for deer about 3-4 weeks ago.
