I've had pretty good success in Montana on the general tag, west of Yellowstone in the Beaverhead and further north and west around Lima Peaks. You can get a general deer/elk combo tag for a little under a thousand and hunt for about five weeks on archery and then about six weeks on the rifle. Some general elk areas are limited entry for deer and vice versa, so you might want to do elk only, depending on where you hunt.

The general elk tag in Wyoming also gives you a lot of good options, especially if you have horses, but that's tough to draw anymore and non-residents need a guide for wilderness areas. Wyoming elk application is in January and you get the results at the end of February, so you'll know before most of the other state deadlines.