How did the women's role in American society change during the period of 1865 to 1940?
During this time, the role of women changed quantitatively but not qualitatively. That is, women came to have a somewhat greater role in society (particularly after 1920) but their role did not change in a major way.
In these years, the main way that women were involved in public life was through participation in reform movements. Women were very strongly involved in the Progressive movement of the first two decades of the 20th century. Of course, they earned the right to vote during this time, which was a major breakthrough. However, the fact that women were participating in reform movements was nothing new. They had been doing this since at least the 1830s when they were involved in such things as the abolitionist movement.
Women did gradually move into a greater role in society during this time. The first woman lawyer was admitted to the bar in this time. A woman argued before the Supreme Court. The first woman was elected to the House of Representatives.
Even so, (and even with the cultural changes of the 1920s) the expected role of women in society did not really change. Women could participate more in economic and political life, but it was still very much expected that their main role would be that of wife and mother.