The .223 or the .308 will both work well on coyotes. I personally use a ruger mini 14 in .223 or 5.56 mm, but that is mainly due to the fact that it is my truck gun and it is with me everytime I go up and work in the back 40. Prior to that, I was using a 1903 Sringfield mark 1 or my Savage 110 in .270 winchester with handloaded .223 sabot rounds. Some people call that overkill, I personally think that overkill is underused.
As some have already posted, .223 will be more cost effective, If I had to choose a new 'yote rifle, I would get a bolt action .223. If I already had a rifle in a good do anything caliber, such as .308, I would run with that. Hope this helps.
If I have to explain why I ride, you wouldn't understand. Explaining sex to an alien would be easier than explaining the feeling, one that boils all that is going on in your life down to what really matters each time you are in The Wind. All I can say is, Life is not Life without it.