How do you protect a Li-ion / Li-polymer battery from over-dishcharge?
The LDO regulator has an inhibit input so use a small voltage comparator looking at the Li-ion battery terminal voltage and when it drops to a certain low level, have the comparator switch so that the LDO Reg turns off via the inhibit input. It's called UVLO (under-voltage lock-out) by quite a few chip suppliers.

You might be able to use that inhibit input directly but it's got a fair amount of hysteresis - to turn the device off requires an input that is below 0.4V and to restore it to full working it seems it needs 1.2 volts.

Also, check that the current draw from the inhibited regulator is low enough not to worry a Li-ion battery that is close to its lower discharge point. If the inhibited current is still fairly significant then you may need to use a MOSFET between battery and regulator.

Or, maybe change your scheme to use a regulator with UVLO built-in: -



The LTC3129 is a buck-boost regulator so even if the input voltage dropped below 3V3 (if that is your output voltage) then 3V3 will continue to be produced. When in UVLO mode, quiescent current is less than 2 uA.