So far I have been controlling the clock by pulsing the pads on the PCB where the crystal oscillator used to be. The ESP8266 board I am using to control the clock sends a square wave into the PCB with a frequency based on the amount of time it is set up to display (a life, a year, a weekend...) It seems to work pretty well, but I think accuracy could be increased by stepping the clock in a different way.

Inside the clock is a coil that when pulsed, steps the internal gearing of the clock. By connecting GPIO pins on the ESP8266 directly to that coil I should be able to control the clock's movement with better accuracy, especially over longer periods of time.