The government's leading economic indicators rose last month for the first time in a year.

The key index, which is supposed to foreshadow changes in the economy, jumped by 2-1/2 percent between May and June, according to preliminary figures released yesterday by the Commerce Department. This was the biggest monthly increase in five years.

Secretary of Commerce Philip M. Klutznick said yesterday, "The strong rise in the index . . . reinforces evidence which has been emerging over the past several weeks that the low point of the recession may be reached sooner than many have predicted."

But he cautioned, "Output and employment are likely to decline further before growth resumes later this year."

The index could be signaling the turning point in the economy from recession to growth, but a Commerce official yesterday said that one month's figures should not be relied upon too heavily.

"If we were to have some more increases next month and the month after," then it would be clear that the country was coming out of recession, he said. "We are not saying the recession is over by any means," he added, but "as Klutznick pointed out, there are other indications that the economic decline is slowing down."

Economists inside and outside the government agree that the worst of the recession is now over, and the government expects an upturn to begin by the end of the year.

Robert Gough, of private forecasters Data Resources Inc., said the one-month figures cannot be dismissed as flukes. "What it says is that the worst may be behind us."

The department yesterday also revised its figures on the May decline in the leading indicators down slightly to 2.3 percent and further reduced the April fall to 3.9 percent. When first published, this fall was put at a record 4.8 percent.

A drop in the rate of layoffs from manufacturing jobs and an increase in building permits led the rise in the leading indicators for June, the Commerce report showed. The layoff rate declined from 3-1/2 percent in May to 2.9 percent in June.

A separate report from Commerce showed that the 0.9 percentage point drop in the layoff rate in June came after three consecutive monthly increases. But the rate is still 1.1 percent above its level of a year earlier.

Another indication that the rise in unemployment is slowing came from the Labor Department yesterday. New unemployment insurance claims dropped by 15,900 in the week ended July 17 from the number in the preceding week, the department reported.

There were 4.2 million people receiving unemployment insurance benefits during the week ended July 12, the department said. This was an increase of 132,900 from the previous week. These weekly figures give an early guide to unemployment figures.

The administration has forecast that unemployment will continue to rise this year to a peak of about 8-1/2 percent.

The new-hire rate reported by the Labor Department was unchanged in June from the May figure of 1.8 percent, the lowest figure since April 1975. jA rise in stock prices was the third-largest contributor to the June increase in the leading indicators index, the Commerce report said.

Improvements also were shown in four other components of the index: a broad measure of the money supply, new orders for manufacturing and consumer goods, contracts and orders for new plants and equipment, and a change in total liquid assets.

Three of the 10 available series which make up the leading indicators declined in June: the average work week for production workers in manufacturing, raw material prices and the length of time vendors took to deliver goods to companies.

Despite the June improvement in the leading indicators, the index -- at 126.9 -- still stands well below its level of a year earlier, when it was 141.6 (the 1967 base equals 100). The June rise pushed the index back to just above its level in April.