Thanks to you all for these very useful replies; has given me some food for thought.
I was not aware that there is little capacity left below even 3.3v, which I had
thought was about middle of usage range [Edit: Brain failure, I do know that the nominal voltage is 3.7]
Open-circuit voltage, aka no-load voltage ranges from about 4.18V (100%) to about 3.4V (0%).
It's just that cells have internal series resistance, so voltage drops under load by U = R*I. Also, this series resistance isn't a fixed value, it grows higher near 0%, so you see more voltage drop. The resistance also increases at cold temperatures, and as the cell ages.
A typical 18650 has internal resistance of approx 50 mOhms, which starts growing to about 100-200 mOhms below about 10-20%. At cold temperatures, this can easily go up to say 500 mOhms, which could be a usable
"not the absolute worst but quite bad case" number.
If you load the cell at 1A, 500mOhms is 0.5V voltage drop so 3.4V no-load voltage becomes 2.9V under-load voltage. But if you only load at 100mA, you are still at 3.35V and not losing almost any capacity with a very low-drop LDO!