The most efficient magnum is relative, relatively speaking.
I have several long range cartridges, some on belted magnums, some on a RUM cases and some on Rigby cases. As far as I can tell, the 338 Win Mag, 340 Weatherby and 330 Dakota all appear to be the most efficient to meet your criteria. Any of these cases with a good high BC bullet will reach 800yrds with ease, even the 250gr Accubond is good for this.
The 300 Win Mag would work, but it is less efficient in powder usage than the 338's.
I run a 338 Edge with 300gr SMK's, would love to try some more Bergers but they are hard to get here. Efficiency is relative, in my books, 3000fps is the goal for any long range cartridge, but sometimes we have to settle for less than this number. My Edge is only 10fps off 3000fps, so I'm happy with it, but it has a 32" barrel to achieve this abd I cannot load to magazine length.
Everything we strive for in this game is a compromise. If you want efficiency above all else, build a 300WSM and don't look back.
Cheers.
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