* Front month still below recent 21-month high * Weather forecasts turn warmer for most of nation * Nuclear plant outages slip below normal By Eileen Houlihan NEW YORK, May 21 (Reuters) - U.S. natural gas futures rose more than 2 percent on Tuesday, extending gains for a third straight session, as warmer weather blanketing much of the nation boosted demand for air conditioning. "The natural gas rally continued for yet another session, not a major surge, but a decent three-day rally which is looking more like new buying is starting to come back into the market after two weeks in a row of the speculative community reducing their net natgas long positions," said Energy Management Institute's Dominick Chirichella. The latest NYMEX natural gas open interest data available showed open interest rose by 4,910 contracts on Monday, the first gain in 13 sessions. Front-month June natural gas futures on the New York Mercantile Exchange rose 10.2 cents, or nearly 2.5 percent, to settle at $4.192 per million British thermal units. 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. Other months also rose on Tuesday, but failed to outpace the front month's gains. Summer months rose less than 10 cents each. In the cash market, gas for Wednesday delivery at the NYMEX benchmark Henry Hub in Louisiana rose 3 cents to $4.13, with late deals firming to even with the June contract, from deals done late Monday at a 4-cent discount. Gas on the Transco pipeline at the New York citygate rose 9 cents to average $4.46. The latest National Weather Service six to 10-day forecast issued Monday called for above-normal temperatures for about the eastern two thirds of the nation and below-normal readings only on the West coast. Nuclear plant outages totaled 15,900 megawatts, or 16 percent of U.S. capacity, down from 16,600 MW out on Monday, 17,300 MW out a year ago and a five-year average outage rate of 16,200 MW. Baker Hughes drilling data last week showed the gas-directed rig count rose by four from the prior week's 18-year low of 350. Early injection estimates for Thursday's gas storage report from the U.S. Energy Information Administration range from 85 billion cubic feet to 99 bcf versus a 75-bcf build during the same week last year and a five-year average rise for that week of 90 bcf.