I do not quite understand the problem. The TPS61021A datasheet says that it will not start until the input voltage reaches 0.8 volts typical so drive the enable pin with a comparator powered from the output to compare the output voltage and battery's voltage to raise the cutoff voltage. Disabling the regulator should then latch it off unless the input voltage rises to 0.8 volts typical.
Isn't it safe to discharge NiMH cells like NiCD cells to zero volts and leave them that way? At least I thought it was. Reverse charging is of course another matter entirely but not a consideration where only one cell is used.
Leave it to Texas Instruments to include a useless schematic; can these guys ever get datasheets (and ADCs, and IC sockets) right? Which side of the PMOS transistor is connected to the output? Is there a body diode? If the regulator is shut off, does the output follow the input minus one Vbe?
Assuming that there is a body diode and the source of the PMOS transistor is connected the output, you will probably need to add a common base bipolar PNP transistor in series with the output to shut off the regulator if you want the output to fall below 0.3 volts; the base connects through a resistor to the regulator's input. The efficiency loss is minimal because the transistors collector to emitter saturation voltage is in series with the output. Check out Linear Technology design note 109 and 110 for an example. There are other micropower boost regulators with built in output disconnects if that is a requirement.