If there were no current in the wire running through the magnet, what could you do to induce an electric current?

If there is no current running through a good conductor, it's always because
there is no potential difference between its ends, so the electrons in it have
no incentive to go anywhere.  If you want a current to flow, just create a potential
difference (a voltage) between the ends of the conductor.  You can do that with
a generator, a battery, a solar cell, or some pieces of metal stuck into a moist
potato.  As soon as you do that, those electrons wake up, and they all start trying
to fly away from the negative end of the conductor, toward the positive end.