The beginning of the fight for womenâs suffrage in the United States, which predates Jeannette Rankin âs entry into Congress by nearly 70 years, grew out of a larger womenâs rights movement.ontinued Challenges. But achieving the right to vote, while ending one phase of the womenâs rights movement, set the stage for the equally arduous process of securing women a measure of power in local and national political office.