The Labor Department on Tuesday reported that a seasonally adjusted 3.8 million people quit their jobs in June, down from 4.1 million in May. That brought the quits rate—the number of people quitting their jobs as a share of overall employment—to 2.4% from May’s 2.6%. In November of 2021, and again in April of last year, the quits rate hit a record of 3%. But June’s quits rate was still a bit higher than the 2019 average of 2.3%, which itself was historically high.

Tuesday’s report showed that the number of unfilled job openings haven’t come down as much. On the last day of June, there were 9.6 million job openings—slightly lower than in May. That left the number of job openings per unemployed person at 1.6—not as high as the record 2.1 notched in May of last year but still well above the 2019 average of 1.2.

Economists generally pay closer attention to openings than quits. Openings lie at the heart of the Beveridge curve—the plotting of the opening rate versus the unemployment rate named after the late British economist William Beveridge that lays out how efficiently the economy is matching jobs with workers. Since the pandemic, the job-openings-to-unemployment ratio has been a particular focus at the Federal Reserve, with policy makers putting forth the idea that a drop in openings could help cool wage inflation without a commensurate jump in the unemployment rate.

But at least in the current instance, quits might provide a more accurate portrayal of job-market tightness. When people quit their job, it is usually because they have a better job elsewhere (job separations for retirement, disability and the like are measured separately). So, as Evercore ISI analysts point out, a quit generally represents an actual bid for a worker. In contrast, openings can represent just options to hire a worker if a dream candidate comes along.

The decline in quits suggests the job market might be closer to the point where it is cool enough for the Fed than what job openings suggest. If that is right, then the Fed might not feel compelled to raise rates again. But in some sectors, quits rates are still quite high compared with before the pandemic. In the leisure and hospitality sector, for example, it stood at 5% in June, which compared with a 2019 average of 4.6%, while in private education and health services it was 2.3% versus 2019’s 1.9%. Both of those are “high-touch” sectors that experienced steep job declines when the pandemic hit, with employers later struggling to hire workers back.

Another way to look at quits rates is to divide them, by total employment, into low-, middle- and high-paying private sectors. The quits rate in the lowest-paying group, which includes leisure and hospitality and retail workers, among others, was 3.7% in June, 0.2 percentage point above its 2019 average. The middle group, which includes education, health and manufacturing workers, at 1.9%, was 0.3 point higher than in 2019. And the highest-paying group, which includes sectors such as finance and information, at 2.2%, was 0.1 point lower.

So the realignment of the labor market that the pandemic kicked off, with lower- and middle-income workers in particular searching for, and finding, better opportunities elsewhere, looks as if it hasn’t played out. The job market might be getting cool enough for the Fed, but that doesn’t mean some businesses won’t still need to dig deep.

Write to Justin Lahart at Justin.Lahart@wsj.com