How does a ferromagnetic object entering a solenoid affect the magnetic field?
A second coil probably is a good way to detect the passing of the projectile. If you have two coils wrapped around the barrel like so:


  coil 1    coil 2
==//////|===/////|===
  |     |   |    |
  +    GND  A    B


A voltage is applied across coil 1 to accelerate the projectile. This acceleration is caused by a change in the magnetic field inside that coil. Let's say it is in the rightward direction. If this is to accelerate the projectile to the right, the projectile's magnetic field must be directed left. As this projectile approaches coil 2 from the left, it causes an increasing, leftward magnetic field in coil 2. This, in turn, induces a voltage in coil 2 that will oppose the changing magnetic field. So there will be a current in coil 2 that creates a rightward magnetic field. Depending on which way you wrap the coil, you will get a positive or negative voltage across coil 2. As the projectile gets farther away, you will see the opposite voltage induced, as the magnetic field of the projectile is still directed to the left, but is decreasing in magnitude. This can be measured with a voltmeter (attach the leads to A and B) just to see the effect. If you want to use that voltage to switch the first coil off, you will need to create a circuit or use a microcontroller such as an Arduino.

Long answer short, in the above configuration, you will see a voltage induced in the secondary coil as the projectile approaches, and the opposite voltage as it gets farther away.