I don't know where you are at but I'm fairly familiar with Utah and if it was only going to be one gun besides my 22 I would think about a light easy to carry quick handling gun. A Savage Scout 10FCM in .308. Iron sights or a low power fixed scope for deer hunting, chances are it would suffice for Elk, but a higher power scope if you must. A lot of Elk has been taken with iron sights and the humble 30-30. There really isn't much better than a good .308. Unless you really like taking the long shots anything bigger isn't really necessary.
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