* Front month still below recent 21-month high * Weather forecasts turning warmer for most of nation By Eileen Houlihan NEW YORK, May 20 (Reuters) - U.S. natural gas futures rose more than 2 percent early on Monday as the return of weekday industrial demand brought with it warmer weather forecasts for the rest of the month. Futures extended Friday's big gains, which came after the Obama administration approved unrestricted natural gas exports from Freeport LNG's Texas terminal, the first liquefied natural gas project to be approved since the start of a heated debate over how best to benefit from the shale boom. As of 9:20 a.m. EDT (1320 GMT), front-month June natural gas futures on the New York Mercantile Exchange were at $4.149 per million British thermal units, up 9.4 cents, or more than 2 percent. The nearby contract hit a one-month low of $3.883 on May 9 after climbing to a 21-month high of $4.444 on May 1. The latest National Weather Service six-to-10-day forecast, issued on Sunday, called for a large swath of above-normal temperatures across the mid-Continent, with below-normal readings only on the East and West coasts. Baker Hughes drilling rig data on Friday showed the gas-directed rig count rose by four from the prior week's 18-year low of 350.