Bob - the small rings, M96's, M94's and/or M98's are or have all been chambered at one time for the 6.5x55, 7x57, 8x57, 9.3x57, 9.3x62, and now, .30/06.
The SAAMi and/or CIP standards for those rounds run a low of 55,000PSI for the 6.5, with the others except for the '06 running 57,000psi. The '06 is 60,000psi.

Thus, normal loading should be just fine.

Many of the old rounds were developed at a time when the smokeless powders were 'flighty' to say the least. They were prone to wide jumps in pressure and were also very sensitive to heat. Thus rounds that might be seen to go to Africa, were loaded very softly. Today, such rounds are still being loaded softly, except for the 7's and 8's on the 57mm case as loaded in Europe by Norma and others. Today, most of the 9's, 9.3's and 9.5's are still loaded quite softly. One can improve upon factory ballistics by a measurable amount.

Since the case capacity is slightly more than the .358 Winchester, it's ballistics should be approachable in your rifles, through careful load development. If not fully confident in this, one could always stay with factory level ballistics.