Specifically, what is the max neck diameter for a 30-06 chamber?
A neighbor fitted his Thompson Encore with a SS MGM barrel but it shot all over the place.

I went to the range with him and noticed there was a large radiused shoulder on the case mouth. I put a micrometer on it and it measured .357" across the neck. We borrowed a fired case from a guy's H&R breakaction and it measured .340" there.
We immediately decided to stop wasting ammo.

Can anyone advise what is the max spec for the neck portion of the 30-06 chamber?

The SAAMI drawing is on line at the Saami site.

The neck is tapered.

At the junction of the shoulder it is .3445 max.
at the case mouth it is .3424.

We got a laugh on a Contender board once.
A guy with a 6mm TCU could not hit anything and there were all sorts of theories about why his new barrel would not shoot. I asked him to check the bore diameter. His 6mm bullet just dropped through it. Turns out they made him a 6.5 TCU.