I would argue a 3 jaw is probably the cause of the runout, you've done pretty well as it is to eliminate that much runout.
Think about it, the jaws are never going to register an absolutely true position on a self centre chuck. The only way to do this is to use a 4 jaw or between centres turning.
Moving the tailstock from left to right will only eliminate any relative bias on that axis. On a self centre fixing, the tailstock can only be used to offer support for an unsupported part, to a decent degree of accuracy, but cannot offer total accuracy.
What makes the conundrum worse is that if you centre drilled the part in the 3 jaw, that will also reflect any corresponding chuck runout when tested on the bar's outer unmachined surface.
Silver steel is a better material than mild to test onto because it's often purchase precision ground.
Michael W
Edited By Michael-w on 14/01/2017 00:42:56