On the vast majority of solid state amps the headphone disconnects the speaker output and powers the 'phones via a series resistor. This means your fault is probably, as you seem to realise, a break somewhere in the output wiring, not in the electronics.
You have a suitable instrument to trace the fault. Put the meter on an a/c range, perhaps 20vac or more.
With one probe at a good ground point, carefully probe from the headphone socket to the speaker leads.