Adding salt makes it colder but it would cause the cooler to reach room temperature slightly faster. The rate at which something heats up depends on the difference in temperature between the object and its surroundings—very cold objects heat up quickly at first, then heating slows down as they approach room temperature. Melting ice uses up heat energy to break the bonds holding the ice together as a solid.

When you add salt, the ice absorbs all the heat energy it already has in order to break those bonds, which causes the temperature to drop below the normal freezing point. The resulting water is colder than the ice it came from and the water conducts heat better than ice, so the water warms up quickly until it gets warmer.

When you don’t add salt, the temperature will pause at 32F/0C until all the ice is melted. When the ice absorbs heat energy from its surroundings, that energy goes to changing the ice into water instead of increasing the temperature. The ice will absorb heat at a slower rate than the super cooled salt water because the ice is warmer than the salt water and because ice doesn’t conduct heat as easily as water does.