* Rewrites throughout, adds U.S. markets open, updates prices, byline, dateline; previous LONDON * U.S. private payrolls increase solidly * U.S. dollar on track for best month in 4-1/2 years * European stocks fall 0.45% NEW YORK, June 30 (Reuters) - Global shares retreated from recent highs on Wednesday, as Asian markets grew jittery about a resurgence of COVID-19 cases and Western markets awaited Friday’s U.S. jobs report and what it might mean for monetary policy. Asset markets have been buoyed over the past year by trillions of dollars of monetary and fiscal stimulus by central banks and governments around the world in response to the pandemic. The success of vaccination rollouts in some places has fueled an economic recovery, and consumer confidence in June surged to 21-year-highs in Europe and 1-1/2-year-highs in the United States. But fears over a sudden rise in inflation and the highly contagious Delta variant combined with investors taking gains as the first half of the year ended on Wednesday. “The search for yield is a very powerful force. It doesn’t have (a) narrative right now to stop,” said Sebastien Galy, senior macro strategist at Nordea Asset Management. MSCI’s all-country world index, which tracks shares across 50 countries, shed 0.16%. It was still set for a fifth straight month of gains, and for a rise of more than 11% in the first half. The Dow Jones Industrial Average rose 132.01 points, or 0.38%, the S&P 500 gained 3.62 points, or 0.08%, and the Nasdaq Composite dropped 22.27 points, or 0.15%. The pan-European STOXX 600 index lost 0.45%. The German DAX fell 116.28, or 0.74%, and London’s FTSE 100 fell 41.47, or 0.59%. EYES ON PAYROLLS Data released on Wednesday showed that U.S. private payrolls increased 692,000 jobs in June, more than expected but less than the 886,000 jobs added in May. That figure pushed the S&P 500 to near record highs on Wednesday. But markets are still focused on the more comprehensive U.S. nonfarm payrolls figures to come on Friday. Economists polled by Reuters were expecting a gain of 690,000 jobs for June, up from 559,000 in May. But the variation among the 63 estimates was large, ranging from 400,000 to more than a million. Yields on longer-dated U.S. Treasuries fell on Wednesday to their lowest levels in more than a week after the private payrolls report. The benchmark 10-year yield, which tumbled to its lowest level since June 21 at 1.448%, was last down 2.9 basis points at 1.4511%. The dollar gained 0.203% on Wednesday, headed for its biggest monthly rise since November 2016. The dollar has gained about 2.6% against a basket of currencies this month, partly in the wake of the U.S. Federal Reserve’s hawkish tilt. A “very optimistic” Fed Governor Christopher Waller on Tuesday said it may need to start dialing down its massive asset purchase program as soon as this year to allow the option of raising interest rates by late 2022. The euro was down 0.25% to $1.1865, while Britain’s pound was last trading at $1.3831, down 0.03% on the day. The Japanese yen weakened 0.30% versus the greenback at 110.83 per dollar. MSCI’s broadest index of Asia-Pacific shares outside Japan closed 0.08% lower, while Japan’s Nikkei lost 0.07%. Chinese blue chips added 0.65%. Indonesia, Malaysia, Thailand and Australia are all battling outbreaks of COVID-19 and tightening restrictions, and Spain and Portugal announced restrictions for unvaccinated British tourists. Oil prices were heading for monthly and quarterly gains after some data suggested U.S. crude stockpiles were shrinking. U.S. crude recently rose 0.97% to $73.69 per barrel and Brent was at $75.22, up 0.62% on the day. Gold headed for its largest monthly decline since November 2016. Spot gold added 0.1% to $1,763.20 an ounce. U.S. gold futures fell 0.07% to $1,761.60 an ounce. Reporting by Carolyn Cohn in London and Elizabeth Dilts Marshall in New York; editing by Jonathan Oatis Our Standards: The Thomson Reuters Trust Principles.