Some 60,000 multifamily units were completed nationwide in June, more than any month in the past 50 years, according to Zillow. That's pushing owners to offer concessions which range from parking privileges to weeks of free rent. Zillow said the share of listings on its platform that offered such concessions topped 33% last month, up from 25% a year earlier. 

"Builders have stepped up and built an incredible number of homes in response to soaring rents during the pandemic, and renters are now seeing the benefits," said Zillow Chief Economist Skylar Olsen. "A slowing job market and lower mortgage rates could mean falling rents if the current trends hold."

Housing has been one of the main contributors to post-pandemic inflation, and further improvement in rental supply should help bring headline rates lower and bolster the Federal Reserve's case for lower interest rates. Nationwide shelter inflation eased to 5.4% in June, from a peak above 8% last year.

In some major metro areas — especially southern and western cities, including Atlanta, Phoenix and Austin — more than half the rental listings on Zillow are offering a concession.

The number of multifamily units under construction hit a record last year. While it's retreated since then, the June figure was still higher than any time since the early 1970s.