There is this thing called stacking tolerances that you have to be aware of. Every single component between the spindle and the work piece has its own run out and the more interfaces you have, the greater the run out will be. If you use that collet block in a 4 jaw you have: the run out of the spindle, the lathe chuck, how accurate you are in dialing in the collet block, the collet block itself, the taper inside the collet block, the accuracy of the collet, the collet nut, how accurately you tighten the collet nut and the accuracy of the work piece. All of these interfaces add up, or stack up, to affect run out.
The most accurate way to reduce this stack is to find/make/buy a direct mount ER chuck that threads onto your spindle. You won't be able to eliminate run out but it is possible to get it waaay down. This is not as critical on the lathe as it is on a milling machine but when working on small precision parts, it can make a difference.
Do you happen to know what the size of your threaded spindle is?