That wasn't actually what I was thinking of. And yes, a 5" chuck is marginally too large for a 6" lathe. However, if you are careful never to run the jaws more than about 3/8" outside of the chuck body, It would allow you to safely chuck a little larger part than you could do with a 4". For example, If you were to chuck a your 4" 3-jaw in a 4" 4-jaw, the jaws might be a little farther out of the chuck body than is good practice. And especially for trying what I was thinking of.
You have been indicating the face of your 3-jaw and apparently finding enough runout that the chuck shows visible wobble when you mount and rotate it. What I had in mind was finding out whether the front of your 3-jaw is parallel to the back, and whether it was parallel to the area around the threads. This would take some time as you would need to make a jaw tension ring to get the 4-jaw's jaws tensioned like they will be with the 3-jaw held in it. Anyway, supposing that you had a 4-jaw, pull the jaws out of your 3-jaw. Turn the jaws in your 4-jaw around so that the tallest part is toward the outside. Temporarily mount the 3-jaw in the 3rd (tallest) step of the 4-jaw and get it roughly centered. Remove the 3-jaw and measure the diameter to the 2nd step. Make a ring of that diameter (you'll probably have to turn the jaws back around). Set the ring against the 2nd step in the 4-jaw's jaws and tighten them. Set up your indicator to read the flats in front of the 3rd step. Set the indicator to zero on jaw #1 and read and record the readings on the other three jaws. Remove the ring and re-mount the 3-jaw in the 4-jaw and center it. Rotate it (if necessary) so that there aren't any screw heads in the way over the spots where you just read the offset of the 4-jaw's jaws. Read and record the readings at the same spots as previously read. If the 3-jaw's back is perfectly parallel to it's front you would get about the same readings as before (if you were using a 5" 3-jaw they would be the same if the 4" is perfect but if you bought a new 3-jaw, I would be inclined to forget your present 3-jaw).. Now move over to the center of the chuck and check the runout on the face that should contact the spindle flange. And check the runout of the unthreaded part down in the hole.
If you actually ever do all of this, at this point leave everything as is and report back.
As justification for buying the 4-jaw, you will eventually need one anyway, regardless of the outcome with your present 3-jaw.
Most things that you can do with the 3-jaw you can also do with the 4-jaw. The 4-jaw does some things that the 3-jaw can't. Like holding square or octagonal stock. About the only thing that you can't with it do is hold hexagonal stock.
