How do I get this spool to work?
The interesting thing with coils (inductors) is current, not voltage. It might be that the circuit connected to your coil uses some type of current-control. In the most simple case, this would be a resistor in series with the coil.

If this was the case, you might measure something like 3.3 V on the connected coil, and you would now measure something like the full battery voltage (7.2 V) on the wires leading to the coil when the coil is not connected.

Also, it may be that the internal resistance of the batteries plays a big role.

Can you try and measure the coil's DC resistance? Can you try and measure the current flowing into the coil? Can you look into the things connected between the battery and the coil?