Kentucky’s Constitution denies the vote to anyone with a felony conviction but allows the governor to restore that right to individuals. In remarks at the State Capitol, Mr. Beshear urged the Legislature to remove that ban through an amendment, but said that he would seek to make the restoration process as automatic as possible until that occurred.

“Our goal is that no one will have to fill out a form,” he said. “And our goal is to create a system and a process where any of these individuals now living in all parts of Kentucky can go into any clerk’s office” and register to vote without difficulty.

“I hope today is just the start of righting a lot of injustices,” he said.

The order signed on Thursday in effect revives a similar order that the governor’s father, Steve Beshear, himself a former governor, signed in his last days in office in 2015. His Republican successor, Matt Bevin, revoked it.

Andy Beshear ended Mr. Bevin’s re-election bid last month, winning the race for governor by barely 5,000 votes. Mr. Bevin, who was widely unpopular even among Republicans, at first claimed the election was marred by irregularities, then said he had lost because Democrats were “so good at harvesting votes in urban communities.”

In Iowa, now the only state with a total ban on voting by former felons, a Democratic governor also signed an order restoring rights to some former felons. But the measure, enacted in 2005, was revoked by his Republican successor. The current Republican governor, Kim Reynolds, has supported re-enfranchising former felons, but the State Legislature has not acted on the matter.

Mr. Beshear’s order won quick praise from civil rights and criminal justice organizations, even as they expressed disappointment that it covered only some citizens who have been returned to society.