An alternative approach would be to use a TL431 + 2 resistors ,
the "reference" terminal will pull the "cathode" = "positive terminal" low when it is above 2.5v ,
so 2 resistors e.g. 7.5k and 2.5k will make it switch at 10v .
The problem with this approach , and many of the simple comparators, is the circuit will likely turn back on again as the battery recovers its voltage, so the 7596 will go on-off-on-off . The 7596 on/off input has a vaguely defined threshold and no hysteresis.
You can add some positive feedback to make it stay switched off, for example by coupling a larger resistor from the output to the reference node, just waving my arms around, I would say 75k (being 10 times the 7.5k , should give 10% hysteresis or 1v in 12v) (150k for 0.5v hysteresis) , this approach might work with the comparator approach too. You will probably need a pull up resistor , the circuit layout would be similar to fig34 in the datasheet. (The TL431 behaves like a NPN transistor/MOSFET but with a 2.5000V threshold )
The fig34 circuit would do what you want anyway, so what's the problem with that? , you can add some hysteresis by attaching a 100k resistor from output to the top of the zener (220k for smaller hysteresis). Getting the zener right for 10v cutoff would be tricky , possibly 9.1V? (The big advantage of the TL431, is you use a trimpot to make an adjustable zener).
Have a look at the TL431 it's a pretty cool part!.