First off I'm assuming you meant 338win mag rather than H&H.
Your 3006 is everything you need for hunting moose and bear. Use a premium bullet in 165 to 180 grain and then go forth and kill an ark full of critters.
As others have mentioned being able to place your shot is more important than the cartridge.
Now if you "want" another rifle I understand and am here to enable you. A 338wm is a wonderful choice for moose and bear. I have shot 338wm for years and love it. It is devastating up close and is effective out to 500 yards with practice.
It is truly the 3006's bigger brother. Similar velocities and trajectories when comparing 180's on the 3006 to 250's in the 338. The lower velocities of the 3006 and the 338 mean less shot meat.
I have a 4570 in an older minty Marlin. The rifle is surprisingly accurate and the kick is mild, however with it's rainbow arc trajectory I view it as a sub 200 yard cartridge.
I do not currently own a fast 30 because of the kick and the meat damage.