I believe what you are referring to is a 'follow-rest' but they are a little tough to incorporate into a taper machine.
What you are experiencing is what would be called 'chatter'.
This can be the result of many causes.
Without seeing your machine & set-up, are you cutting 'shovel' or 'climbing'? This can be determined by direction of travel and the relationship of the bit to the work.
Are you cutting with the face of the tool or it's side?
How sharp is the bit and how tight is the machine?
I wish the answer could be made more simple.
Tapering should be done slowly. Make it a 6-pack.[/QUOTE]
Yeah, but then he will need someone else to come turn the lathe off and load another shaft. If he does it he might get the second shaft on by himself but probably not the third or fourth one.
On the serious side. The lathe RPMs may just be turned up too high if you are using a router. If you are not using a router then that is where to start. Also what rpm speed are you running the lathe and what type of router and bit are you using? How many minutes does it take to do a full pass down the cue? How many thousandths are you taking off at a time? Are you saying that the taper is off a couple of thousands from what you offset the tailstock? If so that is normal. Just adjust it over a little more.