Among the many pleasant reminiscences of these days was the interest manifested by men of prominence in the National game. Among those to be recalled was that grim old warrior General W. T. Sherman, who was an habitual attendant at the prominent games during the years that he resided in St. Louis. Noticing this fact C. Orrick Bishop, then Secretary of the Union Club, and now Assistant Circuit Attorney of this city, caused the General's election as an honorary member of the club and received from the General a characteristic letter of thanks for the honor thus conferred. General Sherman's usual companion on these occasions was Colonel A. R. Easton, after whom Easton Avenue was named, and very frequently Gerald B. Allen, the millionaire foundry man, made one of the General's party of deeply interested spectators.-E. H. Tobias, writing in The Sporting News, December 14, 1895
Best I can tell, this must have taken place sometime between 1874 and 1876 when Sherman's headquarters were located in St. Louis. Tobias includes this story during his detailing of the 1869 and 1870 seasons but the event obviously took place several years later. W. E. Kelsoe notes Sherman's attendance at a game in 1875 and places it in the context of the Brown Stockings' first victory over the White Stockings in May of 1875. With that information, I think it's a good guess that Sherman was "drafted" into the Union Club in 1875.