The components are all soldered and the boards are ready. 
I first soldered all the smaller components leaving display, supercapacitor and solar panels for later. For these, I used solder paste and a hot surface.
Then I programmed the ATtiny10. This way, if the chip was not correctly soldered I could reflow it without removing other components.
As always happen when I successfully solder DFN packages, I've been surprised to see the program being uploaded correctly.
Next, I soldered the display. This seemed daunting at first but ended up being not really troublesome: just use a little solder and a lot of flux.
Supercapacitor and solar panels followed.

I built four boards out of the 10 ordered and all of them work. In the process, I had to throw away a step-up converter: if you solder it reversed you get some magic smoke...
The only missing part is a reset button but this not strictly necessary. I will add it later.
As expected, the board with the IXOLAR panels takes longer to charge the capacitor. 
The one with the larger panel can run an indefinite time when directly exposed to sunlight.
I also found out that a 10uH inductor helps lowering the boost converter start-up voltage.
In future, I'm going to replace the MCP1640 with an IC specific for solar energy harvesting. Another improvement will be replacing the pushbutton with a capacitive pad and dedicated IC