WASHINGTON—The Federal Reserve says there’s evidence that hiring has picked up in recent weeks, even as the job market remains badly damaged by the pandemic. In its semi-annual monetary policy report released on Feb. 19, the Fed said it’s been watching job data compiled by payroll processor ADP. The Fed has constructed its own measurement of hiring using the ADP data, and said that gauge has closely matched the government’s monthly jobs reports throughout the pandemic. “The ADP data indicate that employment improved modestly through early February,” the Fed’s report said. It also said that its measure shows that the battered leisure and hospitality industry—which includes restaurants, bars, hotels, and entertainment venues—has started adding jobs again, after a “temporary downturn” at the end of last year. The Fed has in the past several years turned increasingly to non-governmental sources of economic data to get a quicker, more timely read on …