Normally a piece between centers is suspended between 2 lathe centers which run in conical holes in the workpiece, so centering such a piece is equivalent to how to bore conical holes right in the middle of a round piece. This is easy if the workpiece will fit inside your spindle bore, just stick the piece nearly all the way in through your 4-jaw, dial it in, and center drill using the tailstock chuck. If your workpiece is too big for your spindle bore, then read this:
(from my personal archives - GWE)
-------------------------------------------------------------------------------- > I want to center drill a large shaft. The shaft is about 24" long and > is from an old machine tool, but the ends were never drilled for centers. > The shaft is too big to pass through my lathe spindle. I can suss out how > to face the shaft off square, lay out and centerpunch a hole as closely > as I can, and I can then chuck one end of the shaft on the last 1/8" of > the jaws and hold the other end in my hand and gently "pick up" the > centerpunch with a center drill in the tailstock chuck. That way would > get me pretty close, but it would not be exact. I could then mount it > on a center in the tailstock and bring up a steady rest and then remove > the tailstock center and using a tiny boring tool, bore the center, but > how do I know the steady rest isn't just a little bit off? >
I have had to do this many times before I had a large lathe. First, I assume you have enough space between centers. Chuck a piece of scrap in your chuck and turn it to the same diameter of your shaft. Then set up your steady rest around that piece of scrap so you have the correct pressure on all 3 legs. Lock the adjustments and move the steady rest to the tail stock end of the bed. Make certain the carriage is moved first. Chuck your shaft on the left and suspend the shaft on the right. In the case where your OD is rough, turn a small area clean close to the chuck. Then turn the shaft around and use the steady in the machined area. You can go back and forth a couple of times and then reset the scrap diameter and then the steady, sneaking up on the problem. Keeping the carriage close to the chuck and using just the tailstock, center drill the end. If the end of the shaft is not square and requires facing, simply position the carriage to the right of the steady and do that first.
** Machine a female cap to slide snuggly over the end of the shaft. Drill a hole in the cap the size of your centerdrill and use the cap as a drill bushing.
** Lotsa good suggestions here. But if you haven't done it yet and have a stout live center this is the method I use. The shaft needs to be round and the face fairly square. A good saw cut is adequate. Put the steady on the ways and clamp it. Put the shaft in the chuck and tighten some. Adjust the steady by eye so that the shaft is fairly well centered. Grab a piece of scrap aluminum and put it against the face of the shaft, while holding it there bring the point of the live center against the aluminum. Tighten a little and then back off the steady a bit and use an indicator on the shaft OD to measure runout. Use a hammer to tap the shaft into true. You may, and probably will, have to tighten the center against the shaft while doing this. Once the shaft is running true bring the steady contacts to bear against the shaft OD. Back off the live center, tighten the chuck well, and drill it. If you need the center to run truer than .001 bore it after center drilling. This method takes longer to read than to do.