I've never hunted with a double rifle, but if I had one made up, I'd get one made for a .375-'06 improved.

I have a custom bolt rig in this calibre. I build the cases up on .35 Whelen brass, load 62 grains of IMR4895 for 2600 fps with a Nosler 260 grain partition. I get about 2400 fps with a 300 grain Hornady. Seems to me it'll do about what the old .375 did in it's heyday.