38/55 would work good on a full size Martini Action. I had for a while a 375 2 1/2" NE. Very accurate and pleasant to shoot. Basically a 303 blown out straight. Case on the .303 is a bit short however. NE brass just fine at 2 1/2".

Down side was using it to hunt with Jacketed bullets. Most .375 bullets are made up for H&H velocities. Not any good for deer. I am inclined to build another and use the lighter bullets designed for the .375 Winchester or just plain old cast bullets.

Probably the best cartridge I have seen used on the small Martini action is the 7-30 Waters. It is a good deer cartridge to start with and fits the small action like it was made for it.