As far as the White's go, a lot of people shoot them with a .503 conical which is pretty loose in the bore. They get there best accuracy when the bore is fouled.
It is my belief that the accuracy is better with a larger conical. I am not contradicting Doc when I say this. He believed in using a conical with a little "windage" so a hunter could load a second shot easily.
His words:
"Experience has shown that the average White designed rifle used in a hunting situation shoots so well with a fouled bore and reloads so much faster without having to clean between shots that cleaning between shots becomes senseless. Most White designed rifles will shoot into less than 4 inches at 100 yards even in the poorest light and from the worst positions. In the best of conditions, they often shoot into two minutes of angle and sometimes less.
This quality performance can be bettered only by using custom sized bullets sized tight enough that cleaning between shots is mandatory in order to get the next bullet down the bore. A hunter might tolerate this difficult loading for the sake of accuracy on the target range, but never in the hunting field.
......
In summary, the best compromise between hunting accuracy and fast loading is the White designed bullet, whether SuperSlug, BuckBuster or PowerPunch, with windage of one thousandth of an inch. A tighter bullet can be somewhat more accurate, but may require cleaning between shots in order to load. A bullet with 1/1000th inch windage loads fast, just a single swoop of the ramrod, and is only slightly less accurate, in the range of an inch or less at 100 yards. Certainly that’s enough to worry about at the target range, but certainly nothing to be unduly concerned over in the hunting field. Consistent grouping and a fast second shot are far more important.
Good Hunting ‘Doc’ White"

Myself, I opted for the tighter fitting conical, I never have done what I was told.