It's a little contrarian, but I don't see a whole lot to choose between the Rigby and a 375. Maximum point blank differences will be negligible: 20ish yards probably. If you're wanting to more or less duplicate what you have, a 375 is a superb round, and the 338 is very effective, but still on the heavier end. ( Not that anything is wrong with that).
For me, since it's more of a classic, I'd pair it with a 300 H&H if you like heavy rifles, or, my personal choice, a 7x57mm Mauser, which was sold as as the .275 Rigby, for brand congruence.
I'm big, and not particularly recoil sensitive, but shooting with a PACT timer doesn't lie. I'm faster with lighter recoiling rounds and weapons, whether rifles or handguns, and have as good or better accuracy. A 7x57 makes for a trim, handy, quick little death ray of a deer rifle, and has been used many times to kill elephant. There many more impressive cartridges, but I am drawn to the flair of older cartridges, and a good shot will kill anything with a good shot. The lighter recoil, easier to carry rifle and other factors make me more likely to perform that shot under pressure than a more tiring to carry, rougher recoiling, and slightly scary rifle. I'm not anybody else, and my limitations aren't necessarily yours.
I like commonality and consistency, so I would be awfully tempted to just hunt with the 416 for everything, but have been told by acquaintances who hunted Africa recently that the hunt is getting more specific, like American hunting, rather than how licenses for animals were filled as one came across them, and never knew if a shot may be at a tiny Thompson' Gazelle or an elephant.
