This might be one of those moments where it makes more sense to sell your rifle to a friend, and buy one in the caliber you want.
Remember also that the diameter of .338's bolt face is larger, so some additonal work, if possible, will be required there.
Hey, look at it as an excuse to buy another rifle. If it were up to me I'd hang onto the .308 simply because its an inexpensive centerfire to shoot, easy on the ears and shoulder, and will do the job on nearly everything in the lower
48.