I second RC's recommendation.
Before Bart. B. began imparting his knowledge here , I was of the opinion that the only advantage the .308 had over the '06 was a different throat design and reduced case capacity to improve efficiency with mid-rate burn powders.
I still believe such. The .30/06 IS the superior hunting cartridge. Not just My opinion. But I've digressed.
Try either the 125gr Sierra PtSpt or the new 125gr Hornady SST. Powder isn't really critical. IMR 4064 is my favorite, but there many new entrants lately, most derivatives of benchrest competition, such as H322, Benchmark, IMR8208, not to overlook AA2520, AA2495, and on and on.
I've seen 3,200 fps with 125's and from 26"bbl, 3,300 is possible. I suggest starting with 52.5gr IMR4064 and work up to 55.0. Accuracy trumps velocity.
Where pelts are valuable, try the Hornady 150gr FMJ. It's giving me sub 2moa out of two different M1's, sub MOA from a b/a sporter.
Inside 500 yds, a 125-150gr bullet will shoot flatter, past 500, a 165gr and up will do better with less wind drift. However, a coyote at 500yds is a tough target. In my neck of the woods, you'll rarely see one past ~300yds due to cover. Most coyotes are killed incidental to other hunting. A shotgun with turkey load kills many!