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.
Is it better to do the right thing for the wrong reason, or to do the wrong thing for the right reason? If you do the wrong thing for the right reason, is it still the wrong thing?
Si vis pacem para bellum.
There are two ways of doing something. Right and again.