Moving a magnet through coils wound in the opposite direction
What you are talking about is called a Helmholtz coil but one side reversed.

The main problem from this is it would be less beneficial than two coils in the same direction because of resistance in the wire. You also have magnetic field losses from the second coil.

Imagine a Helmholtz coil with a magnet between them both producing voltage in the same direction, as the magnet moves from one to the other they would produce double the voltage.

If we turn one of the coils around as the magnet generates voltage in the first coil, the second coil is turning into an electromagnet, but not all of it's field goes into pushing the first magnet, because you no longer approximate a helmholtz coil, and the field lines would push against each other and leak out (kind of like pushing the same ends of a magnet together). The other bad part is a small amount of energy will also go into heating the wire in between the coils, which happens in both cases, but further increases your loss because of the field leakage.  



So, if your going to do this, use a helmholtz coil.