If you are looking for a proper varminter you will want a smallbore, high-velocity rig.The only FORMER MILITARY rifles I can think of that fill this would be the 5mm Mondragon semi-auto or the Lee 6mm Straight-Pull. Alas, both of these are just a touch on the pricey side. You might pick up a 5mm Mondragon in the $40,000 price range, then all you have to wory about is magazine limits with semi-autos and hope that you didn't get one of the selective-fire variants. The Lee Straight-Pull, on the other hand, is much more common but the special 6mm ammo has been out of production for 75 years now. And there is always the Meuniere A6, but the ammo for those is a bit scarce..... as are the rifles.
Of the MODERN military stuff, the only one even to think about would be some kind of M-16/AR-15 variant as it is the only one firing a varmint-type cartridge. FORGET AKs, SKSs, CZ858s and things lke that: the ammo just is not up to the job. Even Elmer Keith called the M-16 "a gopher gun" and Elmer was the man who did it ALL, at least according to Elmer.
Most of the guys I know who are using military surplus rifles for coyotes are using either the 6.5x55 Swedes which have been suggested, Lee-Enfields with 150-grain bullets, or converted P-'17s with BIIIG scopes.
If I wanted to do the same thing myself, I think I would opt for a P-'17 in original military trim, fitted out with an S&K Insta-Mount and a Tasco 4 - 12 scope, shooting 150-grain flatbase bullets. That should give you the best of all worlds,even if it did weigh 12 pounds.
Have fun.
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