I have used the 338/06 for years on everything from coyotes to deer to wild hogs to elk, it is an awesome hunting caliber. Lately I have been using Nosler 180 Ballistic tips in it. The Magnum crowd complain about the explosive effect of ballistics tips but at the slower velocities of the 338/06 they are perfect

I used the 338/06's little brother, the 338/308 on all those critters you named. I used the Nosler 200 gr Ballistic tip bullet at 2490 chronographed fps to take White Tailed GNU, Blesbok, Springbok, Impala, Mt Reedbuck, Zebra and Monkeys

Every animal dropped in its tracks except the Mt Reedbuck. As you can see from the picture it was hit a little far back. It just stood there and looked around for me until it fell over.

All the others were shoulder shot per the PH's directions. All bullets exited. (Except the Zebra) All the animals had large bleeding but not bloodshot 1" to 1 1/2" exit holes. (Again except the Zebra) You can see the blood still coming out of the exit hole on the Impala. I call them holes because the didn't look like high velocity bullet wounds but drilled holes. Flesh, bone tissue and hide were absent, but the meat was not blood shot.

Nosler Ballistics tips are the perfect bullet for these slower calibers. And you can't go wrong using them in the 338/06 to hunt those animals.