Here’s the construction, cozily attached to the back of the solar cell:
Same solar cell, I think it can supply up to 4.5V @ 1 mA in full sunlight.
The tricky bit is that the rechargable lithium cell needs to be treated with care. For maximum life, it needs to be hooked up to a voltage source between 2.8V and 3.2V, and the charge current has to be limited.
Note that the 1 kΩ resistor is put in series with the battery not only to charge it, but also when taking charge out of it. Seems odd, but that’s the way the datasheet and examples show it. Then again, with a 10 µA current draw the voltage drop and losses are only about 10 mV. A diode bypass could be added later, if necessary.
The diode after the regulator has the nice effect of dropping the 3.3V output to an appropriate value, as well as blocking all reverse current flow. There is no further circuitry for the regulator, since I don’t really care what it does when there is too much or too little power coming from the solar cell. Let’s assume it’s stable without caps.
It all looks a bit wasteful, i.e. linearly regulating the incoming voltage straight down to 3.3V regardless of PV output levels and discarding the excess. But given that this little 3V @ 3.4 mAH battery has already supported a few days of running time when fully charged, maybe it’s still ok.
I’ll let this charge for a day or two.