Could Mulally be invited to run another huge corporation like Microsoft (MSFT), a possibility that surfaced last year before Mulally said he would stay at Ford (F)? Might the next U.S. President ask him to accept a cabinet office like Treasury, Labor, or Commerce? Maybe a university or business school is looking for a No. 1 with near-magical teambuilding skills.

Mulally, in typically discreet fashion, isn’t ruling anything in or out. In an interview with Fortune, he said he was “amazed at how supportive people have been” and by “the number of opportunities there are to serve and to lead.” At age 68, he remains trim and youthfully vigorous, not to mention independently wealthy. A conventional retirement isn’t out of the question. But he’s also indicated openness to see what’s available and to listen to pitches. Head hunters and search committees are certain to test his willingness to tackle something important, an unusual or tricky challenge that calls for a maestro.

MORE: What Mark Fields faces at Ford

“My vote is for him to give back,” said Noel Tichy, professor of management at the University of Michigan. Society could benefit, he said, from the rare skills that accomplished CEOs employ to solve commercial mysteries such as whether to change brand names, how to develop and sell luxury sedans, and when to borrow and repay capital.

Mulally’s most remarkable skill at Ford was changing the culture. Leading by example, he persuaded and motivated senior managers and executives to collaborate, to trust one another, and to report bad news without fear of repercussions. Amazingly, he accomplished cultural reform without wholesale firings or, in fact, much turmoil at all. Mulally, who joined Ford as CEO in 2006 after a career at Boeing (BA), is credited with leading the automaker through the disastrous crisis years of 2008-2009 mostly unscathed, and in far better shape than domestic rivals Chrysler and General Motors (GM).

Mark Fields, Ford’s new chief executive designee, is slated to take over on July 1. Fields has been running daily operations for about 18 months as chief operating officer. Mulally said last week that he has “nothing left to teach Mark.” Significantly, Mulally will leave Ford’s board, so as not to distract or overshadow Fields’s tenure.

Bill Ford Jr., the automaker’s executive chairman, a few years ago said half-jokingly that “Alan could stay as long as he wants,” so rare and unusual were his executive skills. But that notion, no matter how attractive it sounds, is unrealistic. Corporations must sustain themselves by training and growing new leaders. Mulally, three years beyond typical retirement age, knows this.

MORE: Ford’s aluminum secret weapon

Most retiring CEOs are pleased to fine-tune their golf games, lie in the sun, spend more time with grandchildren, and join a few corporate boards. General Electric’s (GE) Jack Welch, a CEO of equivalent prominence to Mulally’s, retired in 2001 and helped to create an academy to train principals for New York public schools. Welch also worked as an adviser to a private equity firm, endowed a business school, and co-wrote a syndicated column with his wife, a former editor of the Harvard Business Review.

Because Mulally’s run at Ford has ended, he’s likely fielding all kinds of offers. Conventional wisdom for a retiring CEO is not to decide on a new project or job too quickly — but also not to disappear from public view too long. In Mulally’s case, the world won’t soon be forgetting him or the monumental feat of leading Ford away from the brink of bankruptcy.