Tuner “The Last Smoker in America” becomes the latest title to give a commercial Off Broadway production a try, with the musical set for a run at midtown’s Westside Theater that begins in July.Musical with book and lyrics by Bill Russell (“Side Show”) and tunes by Peter Melnick (“Adrift in Macao”) played a developmental run at the New York Musical Theater Festival in 2009 and preemed in a full production in 2010 at Columbus, Ohio’s Catco. Andy Sandberg directs the Off Broadway incarnation, with choreography by AC Ciulla. Set in a world where smoking has been outlawed, “Last Smoker” centers on a family led by a woman who’s trying unsuccessfully to quit. The run at midtown’s Westside Theater puts “Last Smoker” on a growing roster of shows to be produced commercially Off Broadway, which in recent years has acquired a rep as a tough producing climate. Play “Tribes” is already running, while “Cock” opens later this week and “Potted Potter” begins perfs this month. Produced by Sandberg, Whitney Hoagland Edwards, Michael Palitz and Stephanie Rosenberg, “Last Smoker” begins previews July 11 and opens Aug. 2.