As I suggested a few posts ago, the regulator doesn't do much for you. Here is a sim of a "white" 3.2V LED discharging a battery. The Battery voltage is the X-axis. Notice the current I(D1). The battery discharge current goes to ~zero naturally when the battery reaches ~2.4V. The LED will be visibly bright while the LED power is between 50mW and ~20mW.
100's of millions of solar powered garden lights cannot be wrong...
100's of millions of solar powered garden lights cannot be wrong...