Whilst its preferable to do all machining without taking a part out of the chuck its common practice to re-chuck parts in an independent 4 Jaw.
With care its easy to centre parts very accurately, but thats really only at the point where the dial indicator is used. Moving the indicator along the length of a part might show that the axis of the part is describing a cone.
Alignment depends on how square or unworn the jaws are, the only adjustment I can think of relies on a small hammer.
I have read many articles and books where a stage of machining calls for rechucking but I don't recall ever reading any warnings or advice given about ensuring concentricity over the length of a part, it seems to be assumed that the jaws just hold the job square. I think that if I needed to make a machine spindle I would find another way of doing it.
Ian P
on putting a little thought to the subject I realise that the