The union legislature had been meeting in New Orleans in the Lyceum Hall (or Liberty Hall) within City Hall (now called Gallier Hall). 1865, a special committee was looking for alternate meeting sites. The Mechanic's Society offered the use of their building, which was located on the site of the old state house, and Governor Wells accepted. The legislature met for the first time in an extra session at the new building, the Mechanics' Institute, on November 23, 1865.
This image is from the Sanborn map of New Orleans 1895-1896 vol. 2, 1895, Sheet 105. The tinted image is the Mechanics' Institute, (in 1895 called Tulane Hall, University of Louisiana). It would be torn down to make way for the Hotel Grunewald annex tower (today known as The Roosevelt).