Anthony Salters, the powerful and well-connected chairman of the Democratic Party in Hillside, N.J., pleaded guilty in January to federal tax fraud. Three months later, Hillside’s school district hired him for a newly created, $80,000 job teaching homebound students.

He was sentenced in August to six months in prison, federal records show. But that did not stop the school district from giving Mr. Salters, 62, a second job: club adviser at a middle school.

There is no possibility that the district was unaware Mr. Salters had admitted to willfully failing to file taxes; his criminal defense lawyer, Raymond Hamlin, is also the school board’s lawyer.

Even in a state known for political patronage and corruption, the post-plea appointments to two taxpayer-funded jobs — and Mr. Salter’s continuing role as Hillside’s Democratic leader despite being in prison — are striking. And at a moment when New Jersey politics are at a crossroads, they highlight the challenge facing those seeking to upend the old ways of wielding power.

“People keep stealing our watch and telling us what time it is,” said Dahlia Vertreese, Hillside’s mayor.

It is not clear if Mr. Salters is being paid while in prison; district officials refused to respond to questions about his job status or salary history. But his post-conviction hiring demonstrated the degree to which political patronage can sideline sensible policy decisions.