By lateral movement, do you mean the centre is moving along the axis of the body, ie telescoping?
If so, you may be able to eliminate this play by unscrewing the plug at the end of the morse taper. This could reveal a screw which retains the shaft in the bearings. Try tightenig it to remove end-play, Better still, dismantle, clean and re-grease.
Hope that helps.
Clive Brown.