Rotating the magnet (where the poles 'pass through?') Or sliding a magnet-rod through a cylinder-coil?
#1 is to add more turns. #2 is to add more turns of fine wire, so the added turns aren't lifted far from the magnet.
Also, for linear-moving magnet (not rotation,) note that a long coil and short stubby magnet won't work very well. If both magnet poles are inside the coil at the same time, then their effects cancel. Even if both poles are close together while the coil diameter is wider ...same problem. To get maximum output volts, just one pole should pass through the coil, followed by just the other pole. The cylinder-coil needs to be the same length as the magnet, or shorter.
Are you forcing the magnet to move, or instead just letting it slide under its own weight+inertia? If forcing the magnet via attached shaft, then you can make the fields much stronger by adding iron parts as a stator, a flux-concentrator. But this also increases the required force applied to the sliding magnet. Iron parts will hang up a freely-sliding magnet. But for a "pushed piston" setup, the piston force can be huge; much larger than the "cogging" or "sticking" effect on the magnet caused by nearby iron.