If you're in a position where you are limited to factory ammunition, then it will likely come down to trial and error. Any 150-180 bullet that you can find that is legal for hunting (an expanding bullet) and isn't strictly designed for target shooting will be acceptable. While I have little experience with a .308 specifically, it's little brother the .243 made Winchester Nosler ballistic silver tips shoot pretty well. Once I started reloading they became virtually impossible to find anywhere. However, it is quite incredible what they do when you lace one behind the shoulder on a white tail or pig. I don't see why they wouldn't do exceptionally well beefed up to a .30 caliber on mule deer.
Alternatively,Winchester Power-Points, Remington Core-lokts, and Hornady Interlocks would prove to be perfectly acceptable and cheaper alternatives to get more practice in.