The wire coming from the tank is not a "hot" wire. It is a signal wire. The hot wire to the gauge comes from the ignition switch usually. You probably know that.
On my 1960 GMC, the gauge goes to E when key is off.
If you are actually saying the gauge never moves from 3/8, onor off, I would add some gas and see what happens. Could be a stuck needle or bad gauge, etc.
