I am puzzled because, given a small initial movement, I can increase the amplitude of the swing's movement by simply changing my posture on the seat at the peak of each swing.
Swings convert potential energy into kinetic energy and back to potential energy with each swing.
The greater the total energy, the larger the amplitude. Changing your posture on the swing changes your centre of mass. If you raise your legs and sit upright towards the bottom of the swing's oscillation then the work done by your muscles to raise your centre of mass will add potential energy. So continual raising and lowering of your centre of mass with every oscillation will increase the amplitude of the swing. This can be demonstrated easily by suspending a weight from a piece of string attached to a support that allows you to raise or lower the weight by pulling on the string.