TALLAHASSEE -- Gov. Rick Scott says he won’t start looking for a new lieutenant governor until the legislative session ends in May, but the guessing game already is raging over a decision that an unpopular leader must get right.

Suddenly, a job that gets little respect could actually matter. The departure of Jennifer Carroll gives Scott a surprise opportunity to pick a running mate who amplifies his emphasis on education and who, unlike Carroll, won’t have to apologize for making inappropriate remarks or face questions about past ties to a suspect veterans charity.

For Scott, finding a perfect partner won’t be easy: Only one-third of Floridians, 32 percent, say he deserves to be re-elected, according to a new Quinnipiac University poll.

When the search begins, Scott’s criteria will include a hard worker with a proven commitment to public education and unquestioned integrity.

“They don’t have to be as pure as the driven snow, but it would be nice if they were like Caesar’s wife — above reproach,” said John “Mac” Stipanovich, who oversaw the selection of two running mates for former Republican Gov. Bob Martinez. “The first requirement of a lieutenant governor is not rocket science: Don’t get the governor in trouble.”

Scott plans to launch the search in May. The timing means that whoever is chosen will quickly hit the campaign trail on Scott’s behalf, and Carroll’s problems make it more likely that the next No. 2 will undergo an intense vetting to avoid future embarrassments.

With Scott focused laser-like on education, and with Republicans having turned off many Hispanics, the most logical choice is a Hispanic with an education background — preferably from a large, urban county with lots of voters.

Sen. Anitere Flores, R-Miami, 36, a mother of two young children who has stressed education in the Legislature, appears to fit the bill. But she sounds less than enthusiastic about the possibility.

“I’m flattered to have my name out there, but I’m really happy in the Senate,” Flores said.

Another prospect is Raquel Regalado, 38, a member of the Miami-Dade School Board and the daughter of Miami Mayor Tomas Regalado.

“Obviously, I’d be honored to be considered, but I haven’t received anything official from the governor’s office,” Regalado said.

A Republican and native of Miami, Regalado was not a fan of Scott in 2010: She supported Scott’s Democratic opponent for governor, Alex Sink.

“I was one of the few Republicans who supported Alex Sink,” Regalado said. “Alex had a clear education platform at that time, and Rick Scott did not.” She also recalled her “concern” when Scott in 2011 proposed an education budget that would have cut school spending statewide by 10 percent.

Regalado comes from what she calls a “media family,” and her father is a former radio talk show host in Miami. She writes columns for El Nuevo Herald and the Huffington Post and hosts a daily radio program on local politics on WWFE 670-AM, “La Poderosa” (The Powerful).

The single parent of a special needs child, Regalado has a daughter with autism. She supports Scott’s proposal to give every teacher a $2,500 across-the-board pay raise and said Scott is rightly postponing the search for a lieutenant governor so he can focus on his priorities in the Legislature.