You have the bolt face to do any caliber there based on the .308 parent case, so you have a few options.
Depends what model it is now to whether I would actually build off of it. If it's a HBV or Stealth, they are usually exceptional shooters so a rebarrel might be enough, as you would have a decent stock already.
If you are only using it as a core action, you have some easier options. I mean easier in that there are much more and easy to aquire parts for a Remington and not as many smiths may be 'Winchester' guys either.
I personally love Winchesters, but my customs are based on Remington or clone actions (PGW is the exception).