Suppose I want to harvest kinetic energy by moving a magnet back and forth through a coil assembly. Faraday flashlights work by this principle, but to my knowledge those usually only have one coil. Could this design be improved by taking two coils wound in opposite directions from one another, wiring them in series, placing the magnet near the midpoint of the coils, and restricting the magnet's position to stay near that point? Or would mutual inductance from the coils cancel out the harvested energy?
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.