Low-power solar cell source using capacitors
I do not think you specify the values for R9 and R10, which could waste 
 significant power at the current levels you are at when the MOSTFETs are on.  

Why do you need MOFSET control over the circuit that goes from the panel to the caps? If the maximum current to solar cell can produce is 3.8mA why not connect the caps directly to an ATTINY IO pin and just set that pin to on to charge the caps? 

Maybe you can ditch the downstream MOSFET also if the load is less than 20mA?  

You also might want to enable the BOD on the ATTINY so it does not accidentally turn on the MOSFETS when the solar voltage starts to sag. 

Is it intentional that the downstream load will get turned off when the solar cell drops below some voltage... even if only temporarily and even if there is plenty of charge left in the caps?

What are you trying to accomplish here anyway?! :) )