Personally, i'd buy a 30-06 and use 180 grain bullets for both!!! I prefer Nosler partitions for the elk, and any 180 grain will work for the coyotes! :>
On elk sized game, you will have the exact load for that job, and useing the same load on coyotes will punch a hole clear through them, killing them with out blowing the hide apart!!!
Another thing, the coyotes will get you some practise for when that elk comes along useing the same gun and load!! It's a great way to learn your rifle and load, to make you an excelent game shot!!
Drilling Man