Growth in single-family home rentals over the past few years has been most significant at the high end, agents said. The second-quarter market report from William Pitt/Julia B. Fee Sotheby’s International Realty cited “an exploding rental market” among high-end properties in the southern part of Fairfield County, in particular.

Data provided by the firm showed that for the first half of 2018, the number of homes that rented for $5,000 or more a month in the county as a whole was up 33 percent, to 268, compared to 2015. That included 61 properties that rented for $10,000 or more, a 35 percent increase over 2015.

In pricey towns like Greenwich, New Canaan, Darien and Westport, homes at the lower end of that rental range tend to be older properties with three or four bedrooms and a decent-size yard. Those going for closer to $10,000 usually have at least five bedrooms, mint-condition amenities and large lots or a beach-area location.

Westchester also showed growth in high-end rentals, but at a slower pace of just over 12 percent, and almost exclusively in the southern end of the county.

“I definitely see an increase in interest in rentals in the Bronxville area,” said Kathleen Collins, an agent with William Pitt. The cost to rent a home in the affluent square-mile village typically runs between $10,000 and $16,000 a month.

Nevertheless, Ms. Collins said she had been approached by more agents than usual this year asking whether properties she had listed for sale might instead be available to rent. In at least two cases where agents negotiated rental agreements with Ms. Collins, the clients were seeking rentals because they were relocating from abroad.