I've had great luck with Crosman Premier Domed pellets for target shooting in my Benjamin pumpers and those will also do just fine for hunting. Spire points and hollow points are often recommended for hunting, but I question whether they are any better than a good domed pellet for hunting. On the other hand, a domed pellet is clearly superior for long range accuracy in my testing. Way out there at 40 and 50 yards, I have never found any hunting style spire point or hollow point pellet to have the accuracy of a domed pellet, no exceptions. The spire points and HPs will do fine out to 30 or 35 yards, but lose ground in accuracy to a domed fleet beyond that.
Note that when buying Crosman pellets, the bulk brown box pellets have been consistently better for me than the ones in the tins, for whatever reason. With the pellets in the tins, now and then I'll get a tin that just won't shoot worth a darn.
Another great pellet for my Benjamin 397s and my first choice has been the heavier Kodiak Match 10.65s. They are more expensive, but a couple of my Benjamin 397s will do well under an inch at 50 yards with them. The Kodiak Match also work great in some of our other air guns, so I usually keep a good supply on hand and often save money by buying them in large quantities.
If this is a new production 397, be sure to check the muzzle end of the barrel for paint over spray inside the bore. This is a common problem on the current production guns. It will affect accuracy, so removing any paint in the bore is worth the effort.
Lastly, keep in mind that your best velocity on these Benjamin and Sheridan pumpers is usually around 5 or 6 pumps.That's the sweet spot for shot to shot velocity consistency and, therefore, accuracy. More than 5 or 6 pumps only gets you small increases in velocity, but will cost you, dearly, in pumping effort. Back in the day with my Sheridan, I used 6 pumps for all my hunting and never felt lack of power was an issue.