What part number of zener are you using? Don't most zeners waste/conduct/consume a little current even before they reach their nominal voltage?
Try using an N-Channel Jfet, as a 0v drop voltage regulator, with a 1-2 diode adjustment jumper on the gate & GND, between the solar cell and super-cap instead of a zener diode shorting out your solar cell. This will also allow support for higher input voltages, though, your max charging current will be limited by the Jfet. You can also use a P-Mosfet with NPN transistor & diodes at it's gate to create a high current 0v regurator, and you can get rid of the series protection diode. This will however not begin to conduct charge from the solar cell to the super cap until the solar-cell is something like 1.2v higher than the charge in the super-cap.