The nation's industrial output slowed further in October, the government reported yesterday, marking the third month in a row that production has continued at a lackluster pace.

New figures from the Federal Reserve Board showed production in factories, mines and utilities up only 0.3 per cent, following an 0.4 per cent rise the previous month. In August, the index fell by 0.4 per cent.

Moreover, analysts said half the new rise stemmed from distortions resulting from labor unrest in the coal industry. There was some rebound from the coal strike of September and buying ahead in expectation of another.

The continued sluggishness was disappointing to the Carter administration. Courtenay M. Slater, the Commerce Department's chief economist, termed the increase "modest," and "not as robust as we would like."

At the same time, Slater asserted that the administration still expects the economy to grow more rapidly in the final three months of the year than it did in the third quarter. Real output then rose at a 3.8 per cent pace.

The Commerce Department economist cited recent strong gains in personal income and retail sales, which she said point to the likelihood of "somewhat faster" growth in the next several months.

However, some private ecnomists were less optimistic. One analyst noted that the sluggish start recorded in October means the quarter will have to be buoyed by heavy Christmas buying to live up to government hopes.

Top administration policy makers still are worried over the possibility of a slowdown in the second half of next year. It is because of this prospect that the President is considering a tax cut early in 1978.

The bulk of the sluggishness in output last month was concentrated in three or four major sectors of the economy, notably nondurable goods, which rose a modest 0.2 per cent.

Output of business equipment rose 0.3 per cent, despite a strike in the aircraft industry, which slowed production of transportation equipment. And production of materials was up 0.3 per cent.

At the same time, there were solid gains in production of durable goods, which jumped 1.2 per cent mainly because of a surge in auto production. Output of automotive products soared 2.2 per cent in October.

Production of intermediate supplies also rose sharply, jumping 1.1 per cent. And production of construction supplies rose by 1.4 per cent. Output of consumer goods was up 0.6 per cent.

Yesterday's figures brought the overall production index to 6.3 per cent above its level of a year ago, marking a moderate gain for an expansion period. The index now stands at 139.1 per cent of its 1967 average.