Sparano said he first entertained the notion of coaching an N.F.L. team in high school, and 24 years after he started coaching, he took over the worst team in pro football. In between, he coached at Division II New Haven and Boston University, returned to New Haven for four years as a head coach, and was fired as part of staff house cleanings in three of his first four stops in the N.F.L.

All the while, Sparano followed one saying that now hangs in the Dolphins’ locker room: the first step to getting somewhere is to decide that you are not going to stay where you are.

“Unorthodox, no question about it,” Sparano said of his path. “From a small school in West Haven, Conn., to a billboard.”

Indeed, his face is plastered on advertising around Miami, accompanied by the phrase “A New Beginning.” Driving back from his vacation home in South Carolina, Sparano said he nearly drove off the road the first time that he saw it.

Next to him on the billboard were Parcells, the Dolphins’ executive vice president for football operations, and Jeff Ireland, the general manager. The Trifecta, as some call the three men around these parts, worked together in Dallas and migrated to Miami.