ALBANY, N.Y. The number of people without jobs in New York increased slightly last month, nudging the unemployment rate to 8.6 percent from 8.5 percent in April and 8.1 percent a year earlier. The state labor department said on June 14 the number of jobless workers grew to 826,000 from 807,000 a month earlier. That as the state added 6,100 private sector jobs in May, bringing the number added since the end of the recession to 337,000. The private sector accounts for 7,321,000 jobs. The jobless rate in the U.S. ticked up to 8.2 percent from 8.1 percent. In New York City, the rate stood at 9.7 in May, up from 9.5. Over the past year, New York showed gains in professional and business services, educational and health services and hospitality and leisure sectors.