It's still the same total energy. You'll lose ice getting down to minus whatever degrees, so while you're colder to start, you also have less ice.

Ignoring all the extra stuff that can happen (e.g. condensation on the outside of the colder cooler dumping extra energy into it, or freezing and making an insulating layer), a sealed ice+salt cooler should hit 1 degree Celcius before a sealed cooler with ice alone would.