On Thursday, 13 postal workers had tested positive; by Friday, that number was 20, including the one from the Bethlehem facility where Mr. Jackson works, according to Mr. Hogrogian. On Sunday, postal officials in Washington said the number was “fewer than 30.”

With a work force of 630,000, those numbers are still relatively small, but they are expected to keep rising in the coming days and weeks. Mr. Jackson and other rank-and-file workers worry that the Postal Service is not doing enough to protect them, and that they could become unwitting carriers of the virus.

“They had no plan and they weren’t proactive at all,” Mr. Jackson said. “It was just crazy to me.”

FedEx and UPS workers are facing similar fears that their warehouses were contaminated or soon will be.

When Mr. Jackson returned to work the night after the meeting, he said, the cones had been removed from the sick worker’s station and the crew was expected to keep going. Another Bethlehem colleague, Sean Craig, said he was continuing to report for work but was worried about his infant son and his 82-year-old mother.