When I was a young hunter, I used the Winchester 180 gr Power Points for deer in my friend's father's BLR in 308 Win on a couple of whitetails (buck and doe); worked great, but the shots were only 40 and 80 yards respectively (both DRT). The cup and core bullets worked great at the lower velocities of the the 20" barrel on that rifle.
I hadn't shot an animal in 37 years with the 308 Win until this fall when I took a caribou at 76 yards with the Federal 175 gr Terminal Ascent ammo. Again, worked well. Animal fell within a couple of yards from where it stood at the shot.
My favourite hunting bullet since its introduction in the early 2000s is the Nosler AccuBond. Accurate, easy to develop good handloads with, and great on-game performance for over 20 years. I will admit that I have been impressed with the on-game performance of the new Terminal Ascent bullets over the past few years, having taken 6 big game animals with 6 shots, ranging from 43-296 yards.
More review of the ballistics of current factory ammunition offerings for the 308 Win also had me try the Nosler 165 gr AccuBond ammo in my 308 Win.
(Unfortunately, my current 308 Win rifle (LH Browning X Bolt All Weather) did not provide MOA or better accuracy with that factory ammo.
I will be using the PH's son's LH Rem 700 in 308 in South Africa next May so wanted to get reacquainted with the cartridge before that upcoming hunt.)
Depending on what you want to hunt, and where you will be hunting, probably the best all-around choice would be the 165 gr bullet that shoots best in your chosen rifle platform.
For lighter big game at short to modest distances, it's hard to go wrong with a cup and core bullet.
For larger big game, a bonded bullet, or a mono metal bullet will provide better on game performance for penetration and retained weight characteristics on thicker skin, and heavier muscle and bone (should the bone be encountered on a less than broadside (quartering) shot or pulled shot).