Way to much gun for coyotes. I'm a fur hunter myself, and I'd strongly recomend something else like a 22 Hornet, 222, or 223, even if it meant waiting till the funds were available.
Contrary to popular belief, the lighter bullets in high powered rifles will tear up an animal a whole lot worse than heavier ones will. If you "must" use a 30-06 for coyotes, I'd suggest using the heaviest, toughest bullets you can find short of FMJ's. A lot of times the tougher bullets, as long as you don't hit a shoulder on exiting, will only leave small holes in the hides.
When you hit these smaller animals with calibers like the 30-06, bullet performance is not really a factor. Just the sheer shock of that large caliber bullet zipping through the chest is plenty enough for quick kills.