Groceteria.com

Later expanded, presumably for the subsequent 3 Boys Market. Now Barbur World Foods. Judging from the cars and the building, I think it's from later than 1936 though;
http://www.barburworldfoods.com/aboutus.htm

Wow, looking at the live local view, this building looks incredibly intact.

Thanks, I'm glad you enjoyed it. Now that you mention it, the pylon that's still on the left corner of the building looks the same one as the one in the old photo, so maybe the building really is from 1936 and was expanded by Piggly Wiggly in the 1950s. The satellite view shows a clear joint between the original structure on the right and the expanded structure, but judging from the look and proportions of the building in the photo, it looks it was expanded by the time that photo was taken.