Everything was looking up for Hillary Clinton. She was riding high in the polls, even seeing an improvement on trustworthiness. She was sitting on $153 million in cash. At 12:37 p.m. Friday, her aides announced that she planned to campaign in Arizona, a state that a Democratic presidential candidate has carried only once since 1948.

Twenty minutes later, October delivered its latest big surprise.

The F.B.I. director’s disclosure to Congress that agents would be reviewing a new trove of emails that appeared pertinent to its investigation into Mrs. Clinton’s private email server — an investigation that had been declared closed — set off a frantic and alarmed scramble inside Mrs. Clinton’s campaign and among her Democratic allies, while Republicans raced to seize the advantage.

In the kind of potential turnabout rarely if ever seen at this late stage of a presidential race, Donald J. Trump exulted in his good fortune. “I think it’s the biggest story since Watergate,” he said in a brief interview, adding, “I think this changes everything.”

He promised to batter Mrs. Clinton as a criminal in the race’s final week and a half. And Republican House and Senate candidates gleefully demanded to know whether their Democratic opponents were sticking by Mrs. Clinton.