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The page that the quotation is taken from. (995? I have the kindle version.) 
Actually, it was their second hall  and fourth home. Here are sources:
Kilroe, Edwin Patrick Saint Tammany and the Origin of the Society of Tammany or Columbian Order in the City of New York (New York: 1913, n.p.) chapter 4, section 5, ""The Wigwam,"" pp. 189-91

Myers, Gustavus The History of Tammany Hall (New York: 1901, self-published):

p. 6 ""The name ""Wigwam"" was given to their meeting-place, and Barden's Tavern was selected as their first home.""

p. 7 ""the Wigwam at Barden's Tavern""

pp. 13-4 The Tammany Society ""shifted its quarters from Barden's Tavern to the ""Long Room,"" a place kept by a sometime Sachem, Abraham or ""Brom"" Martling, at the corner of Nassau and Spruce streets. This Wigwam was a forlorn, one-story wooden building attached to Martling's Tavern, near, or partly overlapping, the spot where subsequently Tammany Hall erected its first building — the present Sun newspaper building.""

pp. 257-8 14th Street building

Perris, William Maps of the City of New York, Vol. 1 (New York: Perris & Browne, 1852), Plate 10   "