Chava,
This is strange. I looked at some '58's in the gallery and they appear to have four headlights. I'm not familiar with these rigs so I don't know if HI beam is all four lamps illuminated, or if there are two dedicated HI beam lamps only.
What I'm driving at is, is it possible to swap the circuits to see if the problems follows the physical wiring when you stick it on the other headlights.
For the HI beams to flicker and go out, the circuit (or ground) must be interrupted. Then for it to go back on, the circuit must be made again. If there is no thermal circuit breaker or switch in the circuit, it would leave only a faulty wire or faulty connection. A loose connection on the HI beam circuit might work when cold, then as it's powered, the heat builds resistance and the circuit fails. When it cools, it works again.
This makes sense to me, but then a lot of things make sense to me...I would look for a loose connection or even a fault in the wire the HI beams. Has it been determined that the headlight switch itself doesn't have a thermal strip for headlight overload protection.
If you can isolate a single wire for that circuit, you would 'jump around' it with a temporary wire from dimmer to some terminal strip on a inner fender or something.
Having the HI beams turn on and turn off like they do screams loose connection to me!
Stuart
