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 potentialdifference (a voltage) between the ends of the conductor.  You can do that witha 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 tryingto fly away from the negative end of the conductor, toward the positive end.