When the board of Equifax convened last week to discuss the company’s response to an enormous data breach, the 10 outside directors concluded that it was time for their hard-driving chief executive to step down.

There was a problem, though. The roster of possible replacements had been depleted by the fallout from the cyberattack, which had compromised the personal information of much of the adult population of the United States.

Some top contenders were considered tainted, according to two people briefed on the board’s deliberations. Three of the company’s senior executives, including the head of its largest division and the chief financial officer, are under scrutiny for selling stock after the breach was discovered but before it had been disclosed to the public.

Equifax spent five more days limping through the crisis before announcing on Tuesday that the chief executive, Richard F. Smith, would retire.