LAFAYETTE, Ind.—The wine list at the White Castle here proposes a thoughtfully balanced varietal selection, from a pétillant Moscato to a quite approachable Merlot.

Jeanette Merritt stopped by one lunchtime for a tasting. Ms. Merritt isn't new to wine; she's the Indiana Wine Grape Council's marketing director. Wine, however, is new to White Castle. Since December, at this one location, the hamburger chain has been pairing some elegant aspirations with its rather unpretentious "sliders."

At...