I don’t know about the whole state, but northern Alabama had U-Totems in the '60s/'70s (possibly still/earlier). To this day, I can’t explain this: the U-Totem just down the street from my Granny’s had beer taps in the wall near the register; I assume you could just fill 'er up and be on your way.
Northern 'bama also had 6-12s.
And not a convenience store, but Piggly Wiggly has always been an amusing name.
In rural parts of Asia small variety stores, of any name, are referred to as the ‘Mama shop’. Because it’s the shop where your Mama is always sending you to fetch things.