OK, let me guess. You're recoil shy and you don't handload?? Or do you???
When someone asks me what "round" to use, I take that to mean which loaded commercial round.
For the past 10-15 years, hunters have been offered some of the best bullets they have ever had access to.
Any of the premium lines, loaded with premium hunting bullets will be just fine performance wise on game such as moose, right out to 350 meters with bullet weights in the 165+ grain weight range. This is the weight the 308Win really shines with. It's about the best all around bullet weight for the efficient little intermediate case.
Now, it's up to you to find the brand your rifle likes, then practice enough with it to shoot it well.
Being recoil shy has likely caused you to develop a flinch. This means you might choke up or tense up in the split second before you pull the trigger. This might include blinking, tensing or even over relaxing. It may cause you to jerk your trigger instead of squeezing it.
If you have developed a flinch, you need to do two things. First, you need to set aside your 308 and get back to a 22rf for awhile. This will settle the flinch problem. This won't happen with only one range trip. Don't push it. There are close range targets available that simulate different distances by perspective. Use the 22rf to practice placing your shots in the kill zone on these targets. Second, buy a "Past" recoil pad. I use one, because I am recoil shy as well. They are cheap, less than the cost of one box of premium ammo. They reduce felt recoil by at least half.
If you are into spending a bit more money, get a Sorbothane or similar recoil pad installed on your rifle's butt. Get the trigger pull distance adjusted as well to fit you with the pad or coat on.
All of this will reduce felt recoil.
You could get a muzzle brake installed as well but that is almost as expensive as a new barrel.
Practice. Lots or at least as much as you can afford.
I don't have to tell you, shooting from the bench is the worst position for felt recoil. You are in a static position, leaning into the butt, while your butt is glued to a chair. Your body can't move to absorb the shock.
When you hunt moose, will you be in a blind???? The reason I ask, is because the bench should only be used to sight in a rifle and some occasional practice. The rest of your shooting should be from different shooting positions, prone, off hand, kneeling, etc.
This is where a shooting stick comes in handy. Again, I use one as well. It doubles as a light, tough walking stick or can be collapsed to fit in a day pack or hung from a belt loop. There are a couple of models available and run around $35 at most. There are also extendable bipods and tripods but they are awkward to set up and carry and don't come into position quickly. Of course, they are heavier and bulkier.
Sorry for the diatribe but I felt you needed more than you were initially asking for.
