The standard M1A rifle produced by Springfield Armory can be brought up to sub-MOA accuracy if you’re willing to invest the time and money, mostly time. Or of course you can send it out then it’s mostly money.
Any sub-MOA 308 is a more than adequate hunting rifle. Maybe a little heavy. In fact if you can get it down to 1.5 MOA I would have no qualms about hunting with it. If you live in Southeast Texas brush and thicket then any “right out of the box” M1A will due just fine for a hunting rifle because you will never see that 200 - 300 yrd shot. In Southeast Texas 3 inches at 100 yrds will work all day long.
As for caliber. let’s face it, people send their little girls out with 223s to hunt deer and they work. Any 308 cartridge is going to be just fine as long as you can be assured of a good shot.
I would heed the advice about proper loadings for the M1A though. The M1A should have no problems with any quality 165 gr. commercial cartridge. Also the quality of the ammo is going to have a direct effect on the accuracy of the rifle. And poor quality ammo risk things like improperly seated primers or inconsistencies in the powder loads which can be dangerous in an M1A, particularly the primer issue.