Oct 8 (Reuters) - The job market recovery for U.S. women faced a setback in September, much as it did about a year ago after many school districts switched to online learning. But the drop-off in the number of women working or looking for a job last month was much smaller than in 2020 as more schools reopened this year. The data, included in the monthly U.S. employment report released by the Labor Department on Friday, shows that women continue to face a lumpier labor market recovery after being disproportionately affected by job losses during the coronavirus pandemic. It also raises a question over whether the reopening of schools, which likely contributed to this year"s improvement from September 2020, is enough on its own to narrow a workforce participation gap between the sexes that widened during the pandemic. About 350,000 women aged 20 and older left the workforce in September and August of this year, while 321,000 men in the same age group came on board. That leaves the labor force for women aged 20 and older down by roughly 2 million from February of 2020, roughly twice the deficit for men in the same age bracket. "Women face tremendous headwinds as they try to reenter the workforce post-pandemic," C. Nicole Mason, president and chief executive officer of the Institute for Women"s Policy Research, said in a statement. "It will take time to see how women navigate what is hopefully a waning pandemic and what that means for their jobs and careers."Still, the report was an improvement from last year, when the labor force for women aged 20 and up dropped by more than 1 million in August and September after many school districts reopened with virtual learning only, requiring many children to study from home. This year, more women are returning to work as hiring resumes in some of the sectors they are overrepresented in and a larger number school districts reopen for in person instruction. But some mothers are still navigating hurdles that make it difficult for them to work, including challenges securing childcare amid a shortage of workers and temporary school closures in some districts with elevated COVID-19 infection rates. The pace of hiring slowed further in September, when the U.S. economy added 194,000 jobs, following a gain of 366,000 jobs in August. Employment in leisure and hospitality, an industry that is disproportionately staffed by women, rose by 74,000 in September. However, hiring at restaurants and bars was little changed for the second straight month. The number of new U.S. coronavirus cases has been declining since mid-September, a shift that is expected to help the labor market heal more smoothly in the coming months.