* Front month still below recent 21-month high * Weather forecasts turn warmer for most of nation * Nuclear plant outages slip back below normal By Eileen Houlihan NEW YORK, May 21 (Reuters) - U.S. natural gas futures rose nearly 2 percent early on Tuesday, extending gains for a third straight session as warmer weather blanketing much of the nation boosted demand for air conditioning. As of 9:23 a.m. EDT (1323 GMT), front-month June natural gas futures on the New York Mercantile Exchange were at $4.16 per million British thermal units, up 7 cents, or near 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 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 87 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.