Determine which piece of the LDO is the problem. In this case
I think it's likely that the problem lies with pass transistor
leakage / subthreshold conduction, requiring the control amp
to swing to the rail (or nonphysicaly past) to choke the IOUT.
FF usually meaning minimum VT and high temp, even more so.
You may have to do better on the gate driver swing, use a
longer channel output device, etc.