Sanity check: If I have a 3.3V LDO such as this one with a 300mV voltage drop and the input voltage drops to 3.4V, the output would be Vin - Vdrop = 3.1V. Correct? I don't see anything about a low-voltage lockout on this LDO.
Typical behavior for LDOs is that if V
IN starts going below V
OUT plus the dropout voltage, the output starts to droop and becomes approximately V
IN minus the dropout until some point where it gives up altogether and what happens then varies. In this case, the regulator has an ENABLE pin that requires 1 volt to turn the regulator on, so presumably it would turn off at that point if it is still going. This series goes down to 1.0V, so it is possible it would continue to conduct reasonably well down to that point, although your load device would probably have quit long ago. Of course you'd want to actually test its operation if any of this was critical.
Edit--looking further, it appears that the dropout voltage would start to rise when the input goes below 2.0V or so. Keep in mind that the 320mV V
DO is the datasheet limit for maximum current of 600mA. Typical dropout voltage is much, much less at lower currents. Unless you are maxing this thing out current-wise, it will probably be nearly as good as you can get without a buck-boost arrangement.