I recently read "Smokestack Lightning" by Lolis Elie. It seems his favorite 'Q joint in Memphis was the Cozy Corner, 745 North Parkway, Memphis TN.
But, the trip he was writing about was done in '93 or '94, so it may no longer be open, or could be very different now.
Take a little road trip north on I-55 to Blytheville AR (70 miles), and go to The Dixie Pig (oldest BBQ joint in AR), 701 N 6th St (US 61), Blytheville AR. Or try the pulled pork sandwiches at The Kream Kastle, 112 N Division St, Bly AR.