I bought a Winchester Model 70 Featherweight .243 back in 1995, just before they introduced the Classic action with the Mauser claw extractor. After I floated the barrel and lightened the trigger pull, this became one of the sweetest little rifles I've ever shot. I took a deer with it two years ago. Never shot a coyote, but my guide whacked one with his .243 as he and I were hauling out a wild boar that I had shot with my .30-06. It was running, and he knocked that coyote into next week, one of the prettiest shots I ever saw.

If your deer aren't huge the .243 would work fine for you. The .25-06 might also be very good, although I have no experience with this round, and .243 ammo is easier to find (in case you do not reload.)