Consider the .375 JRS. Created by its name sake, gun writer John R. Sundra, it is the 8mm Rem Magnum necked up to .375. The 8mm Rem Magnum is the standard belted magnum case of .375 H&H length. It has a straighter case, sharper shoulder and greater capacity for another 200 fps. or so. This means 2900 fps with the 270 grain bullet and about 2700 with the 300 grain bullet. Ballistically, it's about the same as the .375 Wby.
Of course you can still shoot regular .375 H&H rounds in the .375 JRS and the .375 Weatherby chamber if, for whatever reason, the need to do so arises. Bill Jordan once referenced another .375 Wildcat named after custom gun making out fit in Oklahoma which would chamber the .375 H&H and the .375 Wby if needed.
One problem with the .375 H&H is not that it's insufficiently powerful, but that it's a relatively tapered shell casing and that it's a little short on capacity especially for the heavier bullets. This sometimes results in powder being compressed in the heavier loads and erractic ignition. Bob Hagel wrote about this in the Wolfe Book, Big Bore Rifles. You would not have this problem in the .375 JRS or any of the other "improved" versions of this round. If I an improved round of whatever configuration, I would consider loading it to conventional .375 H&H ballistics with the effect of more uniform ignition and lower pressures.
The .375-.338 (.338 Win Mag necked up to .375 also called the ".375 Taylor") is also very tempting, although you can't use any factory loads in it. If you have a long action so that you can long throat the bullets, the straight .338 case has almost the same capacity as the longer, but more tapered .375 H&H casing and will easily duplicate its factory ballistics. Why not just use the standard .375 H&H if you have a long action? Well the shorter casing is more widely available and is a more efficient shape.