I recently had my 9.3x57 rechambered to the x63 version. I find Hornady dies are best when necking up brass as they come with a tapered expander! Other expander balls are also available from Midway if you want to do an intermediate size up, which I recommend!
What I do is anneal .30-06 cases first, then neck up in stages using Imperial sizing die wax inside the case necks.
I neck up with a .375 neck expander button, then full length size down for a tight chamber fit with the 9.3 dies. This creates a false shoulder to headspace on.
The .30-06 cases are too long to chamber in the 9.3 so I trimmed the cases down to the MAX listed case length.
The cases were then fire formed to the chamber when they shortened, surprisingly to the correct trim length, but MAY still need another trimming.
Yes the .30-06 cases are smaller in the base than 9.3 brass but you are only expanding the bases out once when you fireform after that the 9.3 cases are sized with the correct base size with the 9.3 dies.
Other cases can be used also---.25-06, .270, .280 R (maybe best as they are longer base to shoulder), .338-06 and .35 Whelen.
Some military cases may be better to use to form as they generally are thicker brass (less capacity-use caution when reloading).

Hip