John wrote: (clip) Any advice is welcome ^^^^^^^^^^ You could turn a cone, large enough to seat in the opening in the pepper mill, and turn the back to seat on the live center in your tailstock. Then, a firm pressure against the wood, with the chuck tightened just snug ought to center it.
That's not the say I would actually do it on my lathe, though. I have a laser permanantly mounted over my lathe. I would use the laser, aimed at the free end of the turning, and rotate the lathe by hand, to locate the centering error. Light blows with a mallet or chisel handle will bring the piece closer to center. Then the laser is reset, and you do it again. When it gets really close, place the laser so it projects a semicircle below the work (half on the wood), and you can detect errors down to a couple of thousandths.