I think you answered an important question. You already have and load for the 8x57. You are "wanting" something different/more. The .338Fed isn't that different than the 8x57. However, the .338/06 is a bit more, and different.
I've got both a .338/06 and a .338MarlinExpress. I've also shot the 8x57 a bit too.
It's going to boil down to what you are looking for. I really like the .338's, and built my .338/06 with the possibility of spending some time in Alaska working/loafing around and the 5+1 capacity of the .338/06 in a MkX mauser was alluring. However, since stumbling on a used Ruger M77MkII "Hawkeye-African" in .375Ruger at a ridiculous low price, it has negated all thoughts on the .338/06 and .338ME. Funny on how things happen.....
If you are debating leaving the gun "as is", I would. But, if you are intent on changing it, you could do little bettter than the .338/06, and certainly could do worse. But, realistically, the 8x57 loaded to potential with a 200gr Nosler Partition will do anything you could ask of it in N.America. Most Grizzly/Browns are less than 1,000lbs. A well placed shot with the 8x57 (or .30/06, ect, ect) will drop them in their tracks. The "need" for something MORE lies with the guides that need something to stop a previously "wounded" bear that didn't get the "well placed shot" to prevent a client getting bitten/eaten resulting a lawsuit/insurance claim that costs $$$$$$.