Also keep in mind the 270 factory rounds are pretty grim for accuracy.
I disagree with that statement. I've known several guys with .270's that don't reload, and they all seem to shoot pretty nice groups with them? Perhaps you've just got a persnickety rifle.. most of us have had at least one of those?
As to the original question.... my first choice would be the .260 Rem, in (if possible) a Savage bolt action. Although I've been reading some pretty good reports on the Ruger American. Any of the smaller cartridges would suit your needs very well... the .243, .257 Roberts, 7-08, .... those types. These can all be had in a short action rifle, if you so desire. In my opinion... the 30-06 is more gun than you need for this "project". It will certainly kill deer, and, I assume, coyotes, but it is simply way more gun than you need for the task. More power, more recoil, all of that. The .270 would be about the max I'd consider for this, and I'm sure it would do the job very well.
Be sure to get a good scope. You don't have to break the bank to do it... a good low-end Leupold, or Burris, Weaver, etc.... That will do you more good than you can imagine. Cheap/poor optics will hurt your chance of success more than just about any other factor.
Good luck with your project....