If you used a very thin gauge wire you could measure the resistance from one end of the coil to the connection point. However, even with 28 gauge wire you're looking at a total of 6 ohms for the entire coil, so reading that with a micro is going to require additional components.
Alternately you can measure the inductance of the coil from one end to the tap, which would vary based on which tap was chosen.
However, you're just building an interface. Does it have to be a real coil, or does it just have to look like a coil?
Were I in your shoes, I'd cut the back side of the coil around the taps and use one of many methods for determining which tap was connected. This could be done with one input per tap, or with an analog input if you added resistors to the coil so the resistance significantly increased between each tap.