Repivoting skills are a must. The only way to become proficient at this is practice; lots of practice. Accuracy to within .001 inch is absolutely necessary if repivoting jobs in clock repair are to be done properly. With the piece in the lathe, perfectly centered, bring the point of the graver SLOWLY to the center of the work as it is spinning. In the instant before the end of the graver contacts the end of the spinning stock, holding your breath is a good way to steady your touch . Of course, using the steady rest as a solid anchor for the graver is a must. In order to make the stock end true center visable; square off the end with a file as it is spinning in the lathe. The resulting circular pattern will appear to come to a point; the point is the true center of the spinning stock. If the true center is struck, there will be a cone shaped indentation in the end of the stock. If there is a tiny protruding dimple in the middle of the cone , then center has not been properly struck. You will have to start over with a flat surface again. You must either have exceptional vision, or learn to use a magnifier. A sharp graver is needed. continue to practice striking center and avoid the small dimple in the middle. When you drill the stock go slowly and use a pivot drill whenever possible. The hole that is drilled should be about .001in. smaller than the stock you intend to use. This is true for most repivoting work in clocks on straight shafts larger than .020in.. Use the lathe and wire chucks to install (drive in) the new pivot. It should not be necessary to strike the pivots end very hard to get the pivot to seat properly.
If possible the hole for the pivot should be twice as deep as the pivot is long. This is not always possible, however. If you are drilling into a shaft that has a gear over where the pivot is located ; ;be very careful not to drill into the gear so as to weaken it. Check this out in advance. In order to drill hard steel properly , you will have to draw the temper on it do the work, and then re-temper the piece when you are done (before you install the new pivot). This is the correct way to fix a bad pivot in almost all cases.
Repivoting may seem impossible when you first start, but it will get easier with practice. There will indeed be times when it may not be practical to repivot. For example: a french clock with the pinion gear as an integral part of the shaft with a bad pivot in the opposite ind: the shaft will be very hard steel , and it will be very difficult to strike center. This principal of repivoting applies to more than just repivoting.