Your best bet will be to send it to E.R.Shaw.
They will rebarrel it, and can contour the barrel to match the contour of your present barrel eliminating the need to do stockwork.
Their work is First Rate.
I have a .257Roberts that was barreled with a Shaw barrel back in 1984. It has since delivered consistent sub-moa accuracy. Even though I allowed the gun to get thoroughly water soaked in late 1993, and the barrel "sweated" over the summer, and I got some rusting and subsequent pitting in the barrel, it will STILL shoot it's favorite load under 3/4" at 100yds, and occasionaly under 5/8", out shooting all my other rifles.
Last time I priced a rebarreling, they were getting about $240.00 including bluing to match the reciever.
The "cheapest" way to get a .338/06 would be to trade your "Springer" (Sprinfield-sporter), for a Interarms MkX, and obtain a MidwayUSA "Adams & Bennet" barrel in .338/06.
I was able to get the barrel for ~60.00, and Gunsmith charged me $100.00 to install, finish ream/headspace, and Matte blue the entire rifle.
I think that you will really like the .338/06. Magnum power on a "standard" budget. Recoil, muzzle blast, and knock-down are on the order of the .300magnums.
This is one of the few instances that I differ in my opinion from John Barsness (Handloader/Rifle Magazine).
Game performance has been a "notch" above the .30/06 in my experience. Short range performance matches my warm loaded .35Rem loads (impressive). Long range performance has been very gratifying.
Bullet selection is limited for this cartridge for optimum performance. Most of the .338 bullets have been "toughened up" since the release of the .338 RUM, and the various other .338-.340magnums, and may not give the desired expansion on smaller species of big game ie: WhiteTail deer.
A good example of this is Nosler's discontinuing the excellent 180gr Ballistic Tip and replacing it with the much more expensive 180gr Accubond.
I've got a "life-time" supply of some '70's vintage 210gr Nosler Partitions I picked up at a near "steal" as they had been sitting on a shop shelf for 20+years. Got them for the 1973 dealer "cost"!!!
They can keep their $1.00ea Accubonds !!!!