Perfect 3 jaw chucks are rare animals. There are so many things that affect runout, that the correct question is "Why does my part run true when rechucked in a 3 jaw?"
However, the backplate for the chuck needs to be checked first of all. Take the chuck off and check for runout. It should be dead perfect, there is no excuse for any eccentricity or wobble of the backplate. If you don't get that part correct, then the error exaggerates the further you get from that critical mounting surface.
The current state of wear of the chuck may also be a source of frustration. If it has had some use, and a part has slipped in it a time or two, there may be some slight scarring or wear at the front of the jaws. Carefully stone off any burrs, or, I usually use a round file for this, just lay it on the inner gripping surface of the jaw, and push the file in and out of the jaw. Let the file 'grab' any built up material that is stuck to the jaws. When any buildup is gone, the file will 'skate' over the hard jaw surface. Stop filing at that point
Even a 3 jaw in good condition will likely not hold the stock exactly the same way after rechucking. The potential for runout after moving the stock and rechucking is actually at least twice as great as the actual runout of the chuck to begin with.So if you can, center drill and do not move the stock.
Another trick is to use a collet chuck if you are gripping on good round surfaces and need to do a lot of secondary operations. Collet chucks are quite a bit more repeatable for positioning.
However, for certain work, I have made do with a 3jaw in the following manner: center drill the one end of the stock. Now, move the part way out of the chuck, so that the chuck only grips maybe 1/4" to 1/2" of stock. Put the part on the tailstock center, and then close the chuck on the other end. Because the grip of the chuck is extremely short, it does not have as much leverage to force the stock to run offcenter on the tailstock. Its almost getting close to running the part between centers, which is the surefire way to make sure that there is no misalignment being imparted to the work by the chuck.
On fussy work on a cnc lathe with a power chuck, I will do as described. Then, I will back out the tailcenter and see how much runout the center hole has. It is usually within .01 or .02" (depending on the job length), so then I will smack the outboard end of the stock to bring it into a true running condition. Otherwise, the part will turn a taper due to flexure of the tailstock.
First you get good, then you get fast. Then grouchiness sets in.
(Note: The opinions expressed in this post are my own and are not necessarily those of CNCzone and its management)