For animals I would shoot at 400 yds with an '06, regardless of the caliber or cartridge, I'd be looking at bullet style, bullet weight and velocity for expansion at the longer ranges I may take a shot at. A 300 WM with a 180 gr cup and core bullet (interlock, corelokt, etc) at 2960 fps mv is going basically the same speed at 400 yds as an '06 shooting a 200 gr AccuBond at 2650 fps mv.
As for the 375 H&H, I can see the fun in stretching the range with higher BC bullets. My questions/comments were more about the accuracy as it sits is not bad for a big bore and moose being such a large target and typically not engaged at 400 yards, so it would be a moot point on the last nth degree of accuracy.
But I would go after the low hanging fruit, as has been said such as bedding, possibly cut the existing barrel down and then see where you are at.