You may have something loose in the front end.

Especially the king pins.
Check to see if there's slop there.
Check as well to see if the kingpin bolts are tight.

Tap the kingpin lock bolts in with a hammer blow.
Cushion the blow by using a soft drift - brass, copper, aluminum etc.
Then tighten the kingpin lock/steering stop nuts to the correct torque.
If you simply tighten the nuts the kingpin locks won't be in as tight as they should be.

DennyW makes some good points about caster.
Especially in a T-Bucket with it's large disparity in size between bigs and littles.

Get the car front end up on jackstands when you check for steering component looseness and while you're there, spin the wheels by hand and see if you have an out of round or bent wheel as well as an out of round tire.
Out of round radials are uncommon, but it does happen.
Out of round is more of a problem with bias-plies.

Were the tires balanced on a dynamic balancer?
If done on a bubble balancer they will be in static balance, but possibly not in dynamic balance.
In other words, you could have a side to side unbalance and not realize it if you used a bubble balancer.

And like techinspector1 says, dial in a little more caster if everything else looks good.
Get yourself a swinging protractor from Sears or other and see how much caster you have.
Measure caster on a level floor.

6 degrees positive is a good figure to shoot for on most hot rods.
(Positive meaning the top of the kingpins lean back.)