Living in the Philip Johnson-designed “Wiley House”—a transparent glass rectangle cantilevered over a stone podium—is a bit like being “up in a treehouse,” says owner Frank Gallipoli. The home’s public spaces, including the kitchen, living room and dining room, are located in the glass box, while the bedrooms and bathrooms are in the podium.

The Midcentury Modern home, built in the 1950s, is located on a roughly 6-acre plot in New Canaan, Conn. Mr. Gallipoli, a commodities trader, purchased the home in 1994, restored it, and...