That's their post breakup HQ - they moved in there in 1929. The Boss Tweed days were in 190 Nassau Street and 141 East 14th Street (demolished)
https://en.wikipedia.org/wiki/Tammany_Hall#Headquarters
190 Nassau Street - https://maps.app.goo.gl/3zjkd2mC6PwAYVB26?g_st=ac