The period of 1870 to 1920 saw the rise of a multitude of ritualistic fraternal organizations, for social purposes, mutual aid, or political or ethnic solidarity, to the point where a large percentage of the adult population was a member of one or more of these groups. I'd like to hear your take on the reasons for their rise and decline and your opinion on their effects on society and government, lasting or otherwise.
I'd recommend checking out "A Nation of Organizers: The Institutional Origins of Civic Voluntarism in the United States" by Theda Skocpol, Marshall Ganz and Ziad Munson in The American Political Science Review Vol. 94, No. 3 (Sep., 2000), pp. 527-546.
The article does not focus solely on fraternal organizations, but it does provide answers to your questions. The authors discuss at length how voluntary organizations mirrored the forms of federal government and influenced US democracy from its beginnings on through the time period with which you are concerned. They, like Robert Putnam, also suggest that the decline in membership in these and other organizations has not been healthy for US democracy and civil society.
I think Skocpol's other work on this topic might also be worth looking at for you (find it in the article's citations list).