Building a new rifle, baised on the CZ Magnum action (.532 bolt face) and a 30" .308 barrel.
Usage will be longer range hunting than my 30-06 is capable of handling.
Looking for chambering suggestions (but I won't consider anything with Weatherby in the name, long story). I'm considering things like the 300H&H AI (40*) or PI (30*), and the 300AMU.
Any other suggestions or recogmendations?

300 RUM would fit and the 30" barrel would let you use it's huge powder capacity. You would probably save some money on dies etc using a standard caliber vs a wildcat, and could still neck down 375 RUM cases if you wanted to turn the necks for an exact fit with your rifles' chamber.

Ok, showing my ignorance here. Is the RUM a major leap (in reality) over the 300Wby/300Jarret size cartridges that I mentioned above? I ask only because I see in the Nosler manual that the RUM only runs some 50fps faster in their test barrels.
BTW I will probably stay with 180gr or less bullets, due to the 1-12 twist of the barrel that I'll be using.

I think the RUM is about the same amount larger in capacity than the 300 wetherby, then the wetherby is to the 300 win mag.

In a 24-26" tube the UM doesn't show much gain, but in a 30" tube, you should be able to get good use out of the larger case.

TG, my 1:10 30" pacnor in 300RUM spits out 216gr clinch rivers at 3208fps with retumbo and SD in the single digits. I only need ~22moa to get to 1000yds and it will positively flatten it's intended target. Yes, I feel the 300RUM is better than the wthby and jarret, but you have to use it as it was intended,...(lot's of slow powder and heavy pills with room for the burn to be complete) Only caveat, R-P brass, but match prep can fix a lot of it.

The .300 H&H is a great old classic, but I don't consider it enough of a jump up ballistically over the .30/'06 to justify it over the old '06. However, the .300 RUM is, and it isn't a Weatherby!!

So. You have an '06. You have an action with a .308"x30" bbl. Sorry to be so boring, but chamber that mutha in 300 Winchester Magnum.
With a 30" tube you could spit a 180g out at almost 330 fps, around what that '06 does with a 110g bullet!
The numbers may be off, but believe this: most of those big cases don't do a damn thing at 26" that the smaller ones can do given a little more burn time. If you're ok with a 30" limb sticking off the end of your rifle, you do NOT need to worry about case capacity. The Win Mag has a TON of load dope available, over-the-counter rounds, cases out the wazo, etc., etc.
Look into it. Save yourself some money and wear on the shoulder cuff. The 300 Mag with a 30" barrel is a mean mamma-jamma.

Bawana-be
I have a 375H&H that's donating it's receiver for this project (Magnum length), and the -06 is wearing a fairly new 25" Shilen #5 that is still holding under .5MOA out to 500yd, so I'll stick with that for my "short range, carry rifle", the new one is for "reaching out to touch something hairy" at longer ranges.

Thank you all, looks like I'll go with the Rum. Sierra techs say I will be able to stabilize the 200MK with no problem, and probably the 220MK also.

JustC, I may hit you (and StubbleJumper) up for some load data once I get it finished.

What twist rate? I bet you can do the 240gr pills if you wanted I really wish that Clinch Rivers were available more readily available,..you would love them. You may want to look at the 210gr berger also.

Have fun,..it really has been an easy cartridge for me to get great accuracy from.

How about a .30-338WM?