What methods did women use to gain a stronger voice in social and political affairs during the Gilded Age?
I am not entirely certain you will find many examples of how women were fighting to gain the vote during the Gilded Age. The time period was defined by a rather cynical approach to politics, in which the philosophical tenets of Big Business wove their way into political action. Voter turnout was over 80% in elections primarily because of the spoils system and the idea that a business mentality of seeking to crush political dissent and voice. The Gilded Age will not be one where political reform and the activation of voice will be present.
Yet, I think that one will find that the response to the Gilded Age in the Progressivism Era will be one in which the use of women to gain the right to vote will be on display. Women understood that suffrage existed in the grassroots movement or approach in which "feet on the ground" represented the only way to achieve social cohesion around the idea of universal suffrage. Women took to the streets and became social crusaders. Along with the fight in the temperance movement from women who experienced first hand the destructive effects of alcoholic consumption, crusaders like Alice Paul and Lucy Burns recognized the need to act socially in order to gain political change. Alice Paul used hunger strikes as well as public protests to gain her end objective of achieving the right to vote. Women's suffrage was an issue that fit under the umbrella of political change and Progressivist identity, inspired by the excesses and corruption in the Gilded Age.
During this time, women, of course, did not have the right to vote. Because of this, they had to turn to other ways of gaining a voice in social and political affairs. The main way in which they did this was through participation in reform movements.
Women participated in a number of reform movements. It was during this time, for example, that Jane Addams started Hull House. Other settlement houses began in other cities as well. Women were active in things like the Salvation Army. They fought for temperance and created the YWCA. They also fought for political rights. During this time, women were pushing for suffrage in the various states. They actually won the vote in four Western states by 1900.
During this time, then, women worked to gain a stronger voice largely through working in reform movements.