I dont yet have a dedicated coyote rifle so I use my deer gun. Right now that is Ruger M77MkII in .270 Win. Previously it was a Remington 742 woodmaster in .243 Win
Buy the target gun you want and it will kill coyotes just fine. With a heavily constructed or even a Remington core-lokt bullet you wont cause a lot of damage on your average yote. They arent very big or tough animals so you wont get a lot of expansion. There are rounds available for the .308 with FMJ too but i wouldnt reccomend using them.
I like to see coyotes that are really quite dead so i use a 110 grain spitzer in my .270 loaded to the max with IMR 4831. Bang....thwack....done.
I love the smell of fresh shotgun in the morning.