Since your labor is free you might want to listen to Firemoth. You might also combine what BP said too. If you could learn about primary extraction, trueing/blueprinting, trigger timing, opening/aligning scope-base holes, installing boltknobs, pinning recoil lugs, etc, etc you will know a hell of a lot about what makes an accurate rifle. If you're just chambering a barrel for a grade and then keeping the rifle, I would just get a better action from the get-go.

I think a Bighorn Origin is a great action at a very reasonable cost With a Hunt's Long Range recoil lug, it will headspace identical to a Bighorn TL3.

I also would never chamber a 338 WinMag. So so many better calibers out there.

At the end of the day,what do you wan't do with this rifle? Hunt? Target? Fun?