You need a Dial Test Indicator (DTI) which is a dial gauge with a small probe that can enter the hole and rest against the existing ID; see the picture.
The DTI does not have much travel so you need to center the plate as good as possible by eye before using the probe. You can do this by having the probe just clear of the hole so you can use it as a reference point as you rotate the chuck by hand.
Once you have it close bring the probe into the hole and move it until it contacts the hole and deflects the needle through about half the travel range. Obviously the DTI has to be mounted in your toolpost for all this.
Slowly rotate the chuck, watch the needle to find the minimum and maximum deflection and then stop the chuck with the jaw closest to the point of maximum deflection aligned with the DTI. Now tighten this jaw a little bit and you will see the dial reading change. Only tighten the jaw a small amount so the needle moves maybe 10 divisions on the dial.
Rotate the chuck again and see if the difference between the minimum and maximum has been reduced. If it hasn't align the opposite jaw to before with the dial and tighten this one; you may need to go back to the first jaw and slightly loosen it.
Keep doing the rotating and tightening or loosening jaws until the needle hardly deflects at all when you rotate the chuck. Now go around giving opposite jaws a final tightening and check that the needle still has very little deflection.
It is not likely you will get zero deflection because a good DTI is very sensitive and will pick up inaccuracies in the original hole and even machine inaccuracies.
An open mind is a virtue...so long as all the common sense has not leaked out.
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