Solar cell powered supercapacitor charger with strange output voltage regulator problem
What is the frequency of the oscillation you observe? 

If it is relatively low frequency, then the drop is probably caused just by energizing the TPS61222 output cap and inductor. You could fix that by adding more hysteresis on the comparator. Seems unlikely to me, however, because the super cap is so large.

If it is highish frequency, it could be caused by inductive and resistive drops in the path between the supercap and the input to the TPS61222 (including the internal resistance of the supercap itself, which is likely considerable). Again, adding more hysteresis on the comparator will help fix it. You can also improve the situation by using a bulk input capacitor right next to the inductor to hold the input to the TPS61222 in the face of the relatively large currents you will see at startup - this will work if you take Simon's recommendation and drive the EN pin of the boost converter instead of a load switch. That recommendation is likely a good one anyways, since it saves you a part.

Note that if you are running a load at the output of the boost, you may observe some sort of "oscillation" any time your load power exceeds the input power from the cell. The solar cell will charge up the super cap until the boost turns on, at which point the power drawn from the SC exceeds that going in and the voltage starts to drop. It continues dropping until the TPS61222 turns off / gets disconnected, and then the process starts over.