In another indication the nation is headed for a period of slower economic growth, the Federal Reserve Board reported yesterday the smallest monthly increase in industrial production since winter.

The 0.3 percent increase in output of the nation's factories, mines and utilities was only half as large as in May and well below the heady 1.5 percent in April and 1.2 percent in March.

The administration has predicted a decline in the economic growth rate in the final half of this year, following a spring boom in which the nation's total production of goods and services grew at an annual rate near 10 percent.

In its revised economic forecast released last week, the White House said it expects growth in the neighborhood of 4 percent for the rest of the year.

William Cox, the Commerce Department's chief economist, said the output figure for last month is "one more observation that adds to the conviction that there will be a sharp slow-down in the rate of economic activity."

He said the June increase in industrial output is consistent with an overall growth rate of about 3.5 to 4 percent predicted for the last half of the year. Industrial output accounts for about one-third of the nation's total output of goods and services, the so-called gross national product.

The economy, rebounding from a fierce winter that closed factories, stymied transportation and held the overall level of economic growth to zero, boomed during the spring.

Results of the strong surge in growth this spring showed up boht in inflation and unemployment. The Labor Department last week reported a sharp decline in the unemployment rate, but also said that prices continued to rise at a fast clip.

Administration officials hope the mounting evidence of slowing growth will convince the Federal Reserve Board to ease up on its tight monetary policy under which short-term interest rates have increased a full percentage point over the last three months.

The Federal Reserve Board, worried that fast economic growth is creating new inflationary pressures, has been raising interest rates in an attempt to reduce spending.

Fed Chairman G. William Miller, whose agency has come under intense administration criticism for its interest rate policies, told Congress this week that he thought the interest rate spiral might be near an end. White House officials worry that if the Fed continued to tighten monetary policy the economy could not sustain the 4 percent growth rate needed to keep unemployment from rising.

The current 5.7 percent unemployment rate is the lowest it has been in nearly four years.

The Federal Reserve said that out-put of consumer goods grew 0.3 percent in June, even though there was a decline in the rate at which automobiles were being assembled. Production of home goods, such as appliances and clothing, rose 0.3 percent in June.

Production of business equipment rose only 0.4 percent in June. It was the third month in a row that business equipment production declined, an indication that business investment may be lower than the administration hopes.

If the economic expansion is to continue, even at the more modest pace now anticipated, companies will have to boost their spending on plant and equipment. Consumer buying, which has fueled most of the three-year-long recovery from the 1974-75 recession, is expected to taper off. Home building is expected to decline too, in part because of rising interest rates.

The Fed said output of basic metals, such as steel, continued to expand at a healthy clip.

The industrial production index stood at 144.3 percent of its 1967 average in June.