The nation's unemployment rate continued its gradual decline in May, with civilian unemployment down a notch to 10.1 percent from 10.2 percent in April, the Labor Department reported yesterday.

The jobless total now has fallen by 0.7 percentage point since its peak last December, as the number of Americans out of work has fallen from 12 million to 11.2 million. The total unemployment rate, including members of the armed forces, dropped 0.1 percentage point to 10 percent last month.

President Reagan's economic adviser, Martin Feldstein, said that the report was "more encouraging news" that showed "May was another solid month of economic recovery."

According to a survey of businesses, the economic recovery underway since the beginning of this year has led to an increase in jobs, the Labor Department said. "Since December, payroll employment has increased by 800,000," Bureau of Labor Statistics Commissioner Janet Norwood said yesterday, adding that "job growth has been especially strong--650,000--over the last two months."

In the D.C. metropolitan area, the unemployment rate for April, the latest month available, fell from 5.4 percent to 4.9 percent, the District Deparatment of Employment Services said yesterday. Story, C1

Nationally during May, payroll employment climbed by 375,000, with particularly large job gains in construction, in the automobile and other durable-goods industries, and in services.

However, a separate survey of employment--based on questions answered by householders rather than businesses--showed little change last month in the number of people with jobs, after a sharp rise in April. This led economist Allen Sinai of Data Resources Inc. to describe yesterday's labor report as somewhat "disappointing."

Sinai pointed out that there has been very little change in recent months in the size of the labor force--those who describe themselves as in the job market, whether currently employed or unemployed. Typically, the labor force rises rapidly in the early days of recovery as people who have been too discouraged to look for jobs during the recession start to come back into the job market.

However, the labor force was actually lower last month than it had been in December, before the recovery began. It is this decline that largely explains the drop in the unemployment rate since December, Sinai said, describing the labor-force figures as "an unanswered puzzle" that means the unemployment rate "looks a lot better than it really is."

National Association of Manufacturers economist Gordon Richards commented that "May's unemployment figures do not take into account the 1.8 million 'discouraged workers' who dropped out of the labor force last year. When those workers re-enter the job markets, unemployment could go up" again.

Norwood pointed out that the labor market improvement "has occurred primarily among white adult men and women." The jobless rate for black workers was 20.6 percent in May compared with 20.8 percent in April. This is still more than twice as high as the 8.9 percent jobless rate for whites. Unemployment among black male teenagers rose from 48 percent to 53.1 percent in May, the Labor report said.

The pickup in the economy also has not yet helped many of those who have been out of work for a long time, the Labor report showed. "Those who have been unemployed for long periods of time generally have more difficulty in finding jobs that the short-term jobless do, Norwood said. "The first out are often the last to be recalled."

Norwood said, however, that "the data for May show continued improvement in the overall employment situation," with "large and pervasive" gains in payroll employment. The unemployment rate for automobile workers "at 14.3 percent in May, was more than 10 percentage points below its November recession peak of 24.9 percent," Norwood said.

Feldstein predicted that unemployment would be "somewhere in the mid-to low-nines" by the end of this year, lower than the administration's April forecast of just under 10 percent by the fourth quarter of 1983.

Norwood pointed out that "factory hours and overtime hours, both of which had risen sharply in April, edged down in May." Average weekly hours were 0.1 percent below the April level but higher than in any other month since June 1981.

The labor report also showed a 0.5 percent increase in the hourly earnings index between April and May.