* Shares take a breather after Tuesday’s run higher * Oil prices extend rally beyond $70 a barrel * Aussie fails to benefit from strong economic data * Graphic: Global asset performance tmsnrt.rs/2yaDPgn * Graphic: World FX rates tmsnrt.rs/2egbfVh LONDON, June 2 (Reuters) - Stock markets hovered near record highs on Wednesday as investors cheered the latest evidence of a sustained rebound in global economies and stronger oil prices lifted energy stocks. The mood was less buoyant than on Tuesday, however, as traders waited for crucial U.S. jobs data on Friday to assess what the increasing evidence of a faster-than-expected economic recovery would mean for central bank policy in the United States and Europe. A strong expansion in U.S. and European factory activity in May had lifted world shares to record highs on Tuesday. The broad Euro STOXX gained 0.22% slightly below Tuesday’s record high. British shares extended their rally with the FTSE 100 up 0.36%, while Germany’s DAX and the French CAC 40 gained marginally. The MSCI world equity index, which tracks shares in 49 countries, was 0.1% lower. Futures pointed to a slight fall on Wall Street at the open. Crude oil prices rallied again after closing above $70 a barrel for the first time in two years, aided by investors wagering that the economic recovery would lift energy demand and that supply would fall behind. Mark Haefele, chief investment officer at UBS, Global Wealth Management, said vaccination rollouts would spur “a return to normal patterns of mobility, supporting energy demand”, while support for prices also came from an OPEC showing discpline about production increases. “We see energy firms as among the main beneficiaries of the broader global reflation trend, along with financials,” he said. While broader stock markets remain close to record highs, the momentum of earlier in the year has ebbed as investors begin to worry a stronger-than-expected rebound from COVID-19 means higher inflation and sooner-than-expected monetary policy tightening. Economies are recovering much faster than anticipated -- data on Wednesday showed Australia’s economy racing ahead last quarter as consumers and businesses spent with abandon, lifting output back above where it was last year before the pandemic. That helped the Australian stock market to its latest record, but couldn’t kick the Australian dollar out of its recent range as the central bank has been stubbornly sticking to its dovish tone. SHRUGGING OFF INFLATION DATA Bond and currency markets were calm as traders wait on data for clues as to the recovery’s progress. The Aussie was last down 0.3% to $0.7730. The euro slipped 0.2% to $1.2194, just shy of recent highs as the dollar bounced off five-month lows against major rivals. While investors have built sizeable short positions against the U.S. dollar, some investors worry about a surprise hawkish tone from the Federal Reserve at its meeting later in June. Benchmark U.S. Treasury 10-year rates were steady on Wednesday at 1.6113%. German benchmark 10-year Bund yields slipped 1 basis point to 0.187% lower but have largely shrugged off HICP data on Tuesday showing euro zone inflation rose to 2% in May -- a sign that markets were confident the European Central Bank would not decide to slow the pace of its bond buys when it meets on June 10. “As the major developed economies continue to reopen from COVID lockdowns the focus on central bank meetings is going to intensify,” MUFG analysts said in a monthly outlook note. They expect the ECB to avoid signalling a slowdown in bond purchases, but think the Fed might confirm that “very initial” discussions on tapering its bond buying have begun. Brent futures added 0.6% to $70.65 per barrel and U.S. West Texas Intermediate crude added 0.56% to $68.10 per barrel, despite the OPEC+ alliance agreeing to hike output in July. Cryptocurrency prices were calm, with Bitcoin rising 1% to $37,127. Additional reporting by Andrew Galbraith in Shanghai and Tom Westbrook in Singapore; Editing by Simon Cameron-Moore Our Standards: The Thomson Reuters Trust Principles.