@Mishka Well, since you ask, I think the EM8500 would be interesting to try: https://www.emmicroelectronic.com/sites/default/files/products/datasheets/8500-ds.pdf
I only recently discovered it, but the datasheet says it can self-start with an input voltage of 300mv and a mere 3 microwatts. Once started, the datasheet says it can continue functioning on as little as 100mv and 1 microwatt. It also claims to include MPPT.
The other one that would be interesting to try would be the EM8900, which can self start with an input voltage as low as 5mv, which is, AFAIK, the lowest of any available commercial chip.
However, for a solar cell as small as your solar bit, it would require extra circuitry to operate it in burst mode, because at 5mv the micro-ampere requirements would likely exceed what your solar bit could deliver on a continuous basis. It's the same issue as with the LTC3108: https://www.openhardware.io/view/732/Extreme-Energy-Harvester
I haven't yet tried a solar cell scavenged from a solar calculator, but that might be interesting to try as well, since presumably those are well designed to work with indoor lighting and have been perfected over decades for that use since the 1970's. Unfortunately, I don't know of any that can be purchased outright instead of as part of a solar calculator, like say the FX-260 or similar. Maybe someone reading this knows of a source?
Though it's cheating, the last option would be to use a long-lived button battery purely to avoid cold-boot scenarios and to manage the collection of real solar power during those times. I imagine that such a battery could be quite small if limited to that type of use. Some such batteries might last as long as 40 years, such as Tadiran. That said, if it were deprived of light for long enough, it would probably run the battery down sooner than desired, so I don't like this option.