The Winchester is just begging to be an OKH.
If a Mauser, then the 333 Jeff.
On an Enfield, the Jeff would be appropriate too.
I have brass, some bullets and dies for the Jeffery if you get serious about this project.
There is a 334 OKH which was based on the 300/375 H&H case but I am thinking it used a .338" bullet.
Or.... You could build a .333 Griffin & Howe Super Magnum on that barrel. That is simply a .334 OKH with a .333" bullet.
There is only one of those rifles that I am aware of and it is in my safe. Maybe I should hold on to those .333" bullets...