The 416 Rigby is a general purpose cartridge which does give better stopping power than the 375 H&H, but to be able to use it better as a general purpose rifle, you will have to reload for it.
The excellent book by Pierre van der Walt, "African Dangerous Game Cartridges" (Safari Press) lists more than 30 loads with 350gr bullets (about 2500-2600ft/s), and more than 40 loads with bullets lighter than 350gr (340 - 300gr. These loads are listed with source references, using more than 15 different powder types.
An even more versatile bullet is the excellent Gerhard Schultz HV ("www.gscustom.co.za") of 245gr loaded to about 3000ft/s, or his 330gr HV loaded to about 2600-2700ft/s. These are pure copper (mono-metal) bullets and can be used on any animal on the American continent. If Africa is also on your list, you may want to consider his FP (flat point) solids of 380gr loaded to about 2500ft/s.
The experienced PH' love the 416 Rigby, but it is by no means a charge stopper, in the same class as the 458 Lott and bigger. Many saw it as ideal for lion.