Yes the second month of Adar was in place already at the time of the Exodus from Egypt. In fact, there is no real connection between the Hebrew calendar and the Gregorian calendar. The second Adar is used to ensure that Pesach always be in the Spring as God commanded in the Torah. The seasons are defined by the solar year whereas the Hebrew months are defined by the lunar year which is eleven days shorter than the solar year. That means every year the Hebrew year begins eleven days earlier in the solar year than it did the year before. If the Hebrew months were left to themselves Pesach would move out of the Spring into the winter after a few years. The second Adar ensures that does not happen.