Coyotes don't respond well to bullets that make big holes. They tend to run like hell after being hit. High velocity bullets that don't exit seem to do best.
Stay away from FMJ bullets. I've known a lot of guys to shoot a lot of coyotes and think they missed because the coyote kept right on running. The bullets just passed right though and the coyote died somewhere under a bush a half a mile away. One guy "missed" 17 coyotes in a row. I gave him some hollow points to try and he killed the next 6 he called in.
My fur buyer just told me to expect $50-75 for my bobcats this year. There's no way I'd risk shooting one with a .308
If you are sure there is no possible way you could swap that rifle off for an appropriate sized fur rifle,( Or even a shotgun. ) You might consider investing in a lot of dental floss and a few good needles.
But be ready for some fun. Coyote hunting
is by far the most challenging and rewarding type of hunting I've ever found.