WASHINGTON—The number of Americans filing applications for new unemployment benefits fell to a new 49-year low for the third straight week, though Hurricane Florence’s effect on the jobs market remains unclear.

Initial jobless claims, a proxy for layoffs across the U.S., fell by 3,000 to a seasonally adjusted 201,000 in the week ended Sept. 15, the Labor Department said Thursday. It was the lowest level since December 1969, and less than the 210,000 claims economists surveyed by The Wall Street Journal expected.

However,...