If you don't mind continuing to use what you are currently using for deer hunting, I would seriously recommend that you look at .223/5.56 instead of .308. It'll be great for coyotes, shoot accurately past 600 yards, and it's a heck of a lot cheaper to feed than a .308 is.
If you want to get good at this, you're going to spend more on ammo than the gun; I know I already have, and I'm not good yet.
With that said, I would recommend a Remington 700 ADL. They can be bought used for ~$400 without a scope, and a SWFA SS fixed power will work nicely for only $300 more. Whenever you are ready to move up in scope, you will lose very little on that one--or more likely, decide it's too nice to get rid of and just move it to a different gun. I would suggest the 6x42 for your situation.
No matter what caliber you go with, pick a used 700 or Savage 10 or 110 way ahead of the cheaper offerings like the Axis, American, MVP, or 783. You will appreciate the aftermarket support.
I would like to build an AR in .223 at some point, hopefully this summer. I'm not ruling out a bolt gun in .223 but would like something bigger for deer/hogs right now.