FWIW, I would not use this procedure, ever. Here's why:
First, as others have pointed out, the reading you got away from the chuck is of little value - there can be enough movement in the way the jaws fit into the chuck and the way the part fits into the jaws to allow some movement out at the end of the part. Consider how much deflection you would get if you were actually trying to turn the part way out there without support from a center! I would guess if you removed and re-chucked the part 5 times, you'd get 5 different readings out at the end.
Second, welding produces stresses in whatever is welded. You might be fortunate enough not to induce enough distortion to mess up the fit of your jaws in the chuck ... or not. In any case, precision fits and welding do not go together in my book. (Yes, you machined the jaws afterwards ... but I'm talking about the sliding surfaces where the jaws fit into the chuck.)
Third, and related to the above, unless your jaws are mild steel, welding the soft jaws onto the original jaws runs the risk of cracking due to the stresses induced by welding. If the jaws are cast iron, you would be very lucky not to crack the jaws; if they are a hard steel, you are risking cracking at the joint. If they are mild steel, you can weld with no concern for cracking (though still concern for distortion) ... but if they are mild steel, then you might as well just bore the jaws without bothering to weld something else on.
Glad it worked out for you, but I would suspect that no professional machinist on this forum would dream of following this procedure. I certainly wouldn't, even as a hobby machinist - I value my tools too much to take this sort of risk, especially when it is totally unnecessary; 4 jaw or replaceable soft jaws for me ...