In budding, the general procedure is to graft the bud, then 3 to 4 weeks later after it is apparent it has taken, begin the forcing process. There are 4 ways the bud can be forced; cutoff, bending, breaking and notching. With fall grafting in temperate climates, that is not done until the following spring.
With bark grafting you do cut the stock off and place a stem on the cut off stump. The problem I have had when trying to regraft a bark graft is that the bark will not slip after the stock has been cut off.
One option for regrafting or grafting in general when the bark will not slip is to use techniques that do not require slipping bark--like chip and cleft grafting.
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Skeet