Are you an experienced shotgunner? With a pump? if so, the Remington 7600 pump would be a real good choice. Also very accurate right out of the box and has a removable magazine, which many hunters like.
All it would need is a hunting scope, say a 2-7 zoom, mounted as low as possible.
I have a number of 308 hunting rifles. I like to fondle the Browning BLR and admire the quaility fit and finish. But I find it a bit light in the barrel and don't like the way it feels when shooting off hand.
I like hunting with bolt actions because I carry them with the bolt handle up. I feel safer that way. I once slipped on the ice and watched the rifle land in front of me. I was looking down the muzzle. I was not too excited because the bolt was up.
As mentioned above, the Rem 788 bolt gun is available used. The stock is hockey stick ugly and the metal work is not well finished. But the rifles are very accurate with just about the fastest lock time available. I have two and my sons used tham as el-cheapo F Class target rifles. One of them got a new stock and is still in use as a F Class loaner. If you are a bolt gun guy and can enjoy a rifle for its excellence in function rather than fancy looks, then a $250 788 would be a good choice.
But the rifle I prefer to take hunting is the Rem 7600 pump. It is handy, points well for me and is extremly accurate. I have a 308, a 35 Whelan and a 35 rem in Rem pump actions and love them.
The 308 pump is fitted with a pressure strain gauge and has been used to test a few thousand rounds of ammo. No sign of wear or loss of accuracy.
Here are pictures of the 788 that were turned into a F Class rifle and a target rifle. The rear locking lugs can be seen. The receiver is solid with just a loading port on the side and a small opening on the bottom for a single stack magazine. This makes the rifle stiffer and easier to bed.
I heard that remington discontinued the rifle because too many were buying it instead of the 700. My only complaint about the rifle is that if ammo is loaded hot it stretches due to the rear locking lugs.