In 1923 the Shrine Building at Front and Monroe was completed and its roof garden floor was occupied by the Shrine Roof Cafe. It was managed by the Peabody and the Washington Syncopators played on opening night in December 1923. Now, this cafe was not actually on the roof of the Shrine but on the top floor with those gorgeous floor to ceiling 20' (?) windows with sweeping views of the city, as well as the most remarkable river view in all of downtown. The fact that the Shrine Roof Cafe was inside explains the December opening date.
Going back even further to 1914 we find Memphis' first rooftop club, the Alaskan Roof Garden atop the Falls Building (1912) at Court and Front where W.C. Handy among others performed. That should have been a cool place.
There was also the rooftop at the Claridge Hotel, now condos, at Main and Adams. Popular in the 1920's and 30's and called at one time the Starlight Roof Garden and also the Cascades Roof.