“I sure hope that doesn’t happen again next year,” Mr. Zemsky said, in a deadpan.

Mr. Cuomo may also be using ride-hailing to lure recalcitrant upstate lawmakers to the table to address other issues. Companies like Uber are able to operate only in New York City because state laws for group insurance policies make it financially impractical to operate elsewhere. Industry backers point to a variety of evidence that upstate residents — and voters — want to use the apps. (In one case, Uber cited internal data to say that 43,000 people tried to gain access to its app in upstate New York on the day before Thanksgiving, “one of the busiest drinking nights of the year,” though that figure — or that drinking — could not be verified.)

Adrian Durbin, director of policy communications for Lyft, another major ride-hail company, noted that ride-sharing laws had been adopted in 36 states. “We are hopeful that Albany will soon decide to bring all those benefits to New York State,” he said in a statement.

Bills to change such insurance restrictions have run into opposition in the last two years from an array of groups, including the taxi industry, advocates of accessibility for the disabled and labor groups concerned about driver wages and the lack of other benefits. Jonathan Westin, the executive director of New York Communities for Change, said that “Uber is quickly becoming one of the largest low-wage employers in New York,” and suggested that the governor, who championed a higher minimum wage this year, should not back a company that helped “exploit and devalue its drivers.”

John Tomassi, the president of the Upstate Transportation Association, which represents taxi and livery car interests, said its drivers simply wanted “a level playing field” with ride-hailing companies, including structured fares and “basic public safety” requirements like fingerprinting.

He said he did not understand the rush to address such changes in a potential special session. “Why wouldn’t this be open to public discussions and hearings?” Mr. Tomassi said. “Why is this so special that we have to ram this through?”

Ride-hailing companies seem willing to compromise, including paying new taxes on rides outside New York City, though they have opposed fingerprinting drivers, who the companies say are often part-time workers making money on the side.

“Given the progressive leadership that the Assembly and the governor have played on criminal justice issues, we would be shocked,” if they embraced fingerprinting, said Josh Gold, an Uber spokesman, saying it should not be “an employment check.”