John,
Stew has hit the nail on the head.
Accuracy always comes at a price. That is a fact of life. Either you have to spend time setting up with a four jaw independant, waste materials to make holding mandrels, that can only be used the once, because when they are taken out of the chuck, they will have to be recut, as they lose all concentricity as soon as you slacken off the chuck jaws. Or you purchase items that will allow you to get the repeatable accuracy that you require.
I am the type who will spend cash to get it, I am not boasting or gloating about it, It is just that I class my time as more precious than cash, and I need it to carry out the jobs I do for monetary exchange. But don't get me wrong, I am not silly about it, if I can get something cheaper to do the job, I will do, I don't go for the most expensive bits, like some people do. But at times, quality does make a huge difference. The cost of machine tooling for say a new lathe, will cost between about 50% to over 100% of the cost of the original machine. Again hard facts of life.
Not everyone has that facility, again, a fact of life, and doing it on a shoestring is nothing to be ashamed of, family and personal needs must always come first. So in that situation, you usually have to do it the long and hard way.
I have found and always said, that the cheapest way to get spot on accuracy for circular parts using a 3 jaw is to invest in a set of soft jaws for your chuck. Mine cost me about 20 to 30 squid a set, and I have them for every self centring chuck I own. If they are used intelligently, they will easily last the life of a chuck (again, I will replace my most used chucks every few years, they do wear over time).
If I get the time over the next few weeks, I will do a bit of a write up to show you how to easily improve your accuracy, either with standard practices, or spending a little cash, to make it a permanent thing.
John