* Front month remains just under recent 21-month high * Weather outlooks mixed for May, some cool weather this week * Coming Up: EIA natgas storage data Thursday By Eileen Houlihan NEW YORK, May 1 (Reuters) - U.S. natural gas futures edged higher early on Wednesday, boosted by some near-term cool weather and expectations for another light weekly inventory build on Thursday. A long cold winter put a huge dent in inventories, and lingering cool weather in the United States this spring has led to a slow start to the injection season. Still, most traders expect the onset of milder spring weather in the coming weeks to finally curb any late-season heating demand before heavy cooling loads kick in. As of 9:19 a.m. EDT (1319 GMT), front-month June natural gas futures on the New York Mercantile Exchange were at $4.397 per million British thermal units, up 5.4 cents, or just over 1 percent. The contract rose as high as $4.429 in late April, its highest mark since late July 2011. The latest National Weather Service six to 10-day forecast issued on Tuesday called for above-normal temperatures for about the western third of the country and in most of the Northeast. It also forecast mostly below-normal readings elsewhere across the Southeast, Texas and much of the middle of the country. ANOTHER LIGHT INVENTORY BUILD Last week's gas storage report from the U.S. Energy Information Administration showed domestic inventories rose the prior week by 30 billion cubic feet, below Reuters poll estimates for a 32 bcf build, the year-ago gain of 43 bcf and the five-year average build of 50 bcf for that week. Inventories also started the injection season about three weeks later than expected due to the unusually cold spring. Stocks, at 1.734 trillion cubic feet, are nearly 32 percent below last year and more than 5 percent below the five-year average. Early injection estimates for Thursday's weekly storage report range from 24 bcf to 35 bcf, versus a 31-bcf build during the same week last year and a five-year average rise of 67 bcf for that week. EIA data on Tuesday showed gross natural gas production in February climbed for the first time in three months. Output rose to about 1.27 bcf per day, or 1.8 percent above the same month last year, after dropping below year-ago levels in January for the first time since 2010. The report dimmed prospects that record high production would slow anytime soon despite Baker Hughes gas drilling rig data last week that showed the rig count dropped to a 14-year low.