Demand grew particularly in Chicago and Houston. Move-ins in those markets outpaced apartment construction by nearly 3 to 1, RealPage data showed.

Monthly rental prices nationwide shot up 3% from the second quarter of 2018 to the same period this year. Many small metros saw bigger increases. Rents rose 7.4% in Wilmington, N.C., and 6.4% in Huntsville, Ala., according to the report.

Although apartment construction is near the highest it has been in 30 years, much of that new supply is targeting higher-income earners, Mr. Willett said.

With little new affordable supply, the market for lower-cost rentals is much tighter. Most of the new supply “doesn’t really help the consumer in [lower-end] housing where it’s already full,” said Mr. Willett.

The volume of existing-home sales on an annualized basis has fallen for 15 straight months. But the asking price of homes is still increasing at a rate that is faster than income growth.

The shallowing buyer pool and the lack of affordable homes has helped fuel the demand for rentals, said Calvin Schnure, economic analyst at the National Association of Real Estate Investment Trusts. “There’s a shortage of every type of housing,” he said, noting construction hasn’t kept pace with population growth.

Last week, the Trump administration announced it would explore using federal programs to reduce local barriers to housing construction, such as restrictive zoning. Freddie Macalso released a survey showing that American renters increasingly believe they will never afford to buy their own home.