Hi, the S terminal at the starter solenoid should not be hot, except with the key in the start position. When you release the key, it springs back to the run position and the S terminal shouldn't be hot?
If no voltage at S terminal, usually purple wire, engine running, the pinion gear may be hung up on the flywheel. Does your starter use shims? With eveything turned off, remove flywheel dust cover, where applicable, see if pinion gear is still engaged to the flywheel?