(CNN) -- The day before she went to the hospital with Ebola symptoms, Amber Vinson was flying halfway across the country on a commercial jet with 132 other people.
Centers for Disease Control and Prevention Director Dr. Tom Frieden said she never should have stepped foot on the flight, but another federal official told CNN that no one at the agency stopped her.
Before flying from Cleveland to Dallas on Monday, Vinson called the CDC to report an elevated temperature of 99.5 Fahrenheit. She informed the agency that she was getting on a plane, the official said, and she wasn't told not to board the aircraft.
After authorities announced the 29-year-old nurse had been diagnosed with Ebola on Wednesday, they were quick to say guidelines weren't followed when she took the commercial flight.
Dallas Mayor Mike Rawlings said he wasn't sure how it happened.
"She was being monitored here in Dallas," he told CNN's "The Situation Room."
"And if she was being monitored correctly, I think she should have never gotten on that flight."
Frieden said Vinson shouldn't have flown because she helped care for Ebola patient Thomas Eric Duncan at Texas Health Presbyterian Dallas, and because another health worker who cared for Duncan had already been diagnosed with the virus.
"The CDC guidance in this setting outlines the need for what is called controlled movement. That can include a charter plane, a car, but it does not include public transport," Frieden said. "We will from this moment forward ensure that no other individual who is being monitored for exposure undergoes travel in any way other than controlled movement."
2 Ohio schools close
Two schools in the Solon School District in suburban Cleveland are closed Thursday as a precaution because a staffer "traveled home from Dallas on Frontier Airlines Tuesday on a different flight, but perhaps the same aircraft," as Vinson, the school district said in a statement.
"Although we believe what the science community and public health officials are telling us about the low risk of possible transmission of the virus through indirect contact, we are nonetheless taking the unusual step of closing the dual school building for Thursday so that we can have the schools cleaned and disinfected," the district posted on Facebook.
The agency also is considering expanding its "no-board" list for people with known infectious diseases to include people being monitored for Ebola, a CDC official said.
CDC guidelines warn airport screeners that travelers returning from West Africa with a temperature of 100.4 Fahrenheit and higher could be showing Ebola symptoms. That threshold is nearly a degree higher than the temperature Vinson reported.
Frieden said that Vinson's slightly elevated temperature was another sign she shouldn't have boarded the aircraft.
Vinson didn't show any symptoms when she got on the flight Frontier Airlines flight from Cleveland to Dallas, the airline said.
Frieden said there's an "extremely low" risk to anyone else on that plane, but the CDC is reaching out to everyone on the flight as part of "extra margins of safety."
Vinson was transferred Wednesday night from the Dallas hospital to Emory University Hospital in Atlanta, which has successfully treated two other patients. It is now treating a third: a male health care worker who was infected in Sierra Leone.
Vinson is "ill but clinically stable," Frieden said.
The first Dallas health care worker diagnosed with Ebola, Nina Pham, remains in good condition, officials said. It has not been determined whether she will be transferred to another facility.
Both Dallas health workers had "extensive contact" with Duncan on September 28-30, when he had "extensive production of body fluids" such as vomit and diarrhea, Frieden told reporters in a conference call.
Health care workers who had been exposed to Duncan were undergoing self-monitoring.
CDC wants to interview passengers
Vinson flew from Dallas-Fort Worth to Cleveland on October 10, Frontier Airlines said. On October 13, she headed home.
That's the flight health officials are concerned about, though they stress that the risk of exposure to passengers who were on the plane with Vinson is low, since she did not yet show symptoms. The Ebola virus is not contagious before symptoms set in.
"Because of the proximity in time between the evening flight and first report of illness the following morning, CDC is reaching out to passengers who flew on Frontier Airlines flight 1143 Cleveland to Dallas/Fort Worth Oct. 13," the CDC said in a statement.
The October 13 flight was cleaned thoroughly after it landed, "per our normal procedures which is consistent with CDC guidelines," the airline said. After the airline was informed of the Ebola patient, the plane was removed from service.
After going through decontamination, the plane was going back into service on Wednesday, Ricky Smith, Cleveland's Director of Port Control, said at a news conference. Both the CDC and the airline were comfortable that it was safe to resume operations, he said.
In a sign of growing concerns about Ebola, President Barack Obama canceled trips to New Jersey and Connecticut on Wednesday to convene a meeting at the White House of Cabinet agencies coordinating the government's response to the outbreak. The President canceled planned travel Thursday in order to oversee the response.
Obama said he called Wednesday's meeting after the second nurse's diagnosis.
"We're going to make sure that something like this is not repeated, and that we are monitoring, supervising, overseeing in a much more aggressive way exactly what's taking place in Dallas ... and making sure that the lessons learned are then transmitted to hospitals all across the country," he said.
Hospital denies 'institutional problem'
The news that Vinson contracted Ebola cast further doubt on the hospital's ability to handle the virus and protect employees.
On Wednesday night, a federal official told CNN Chief Medical Correspondent Dr. Sanjay Gupta that staffing issues at the hospital were behind the decision to transfer Vinson to Emory.
"What we're hearing is that they are worried about staffing issues and a possible walkout of nurses," the official said.
Some nurses at the hospital have slammed conditions there, telling a union they felt "unsupported, unprepared, lied to and deserted."
It's the same hospital that initially sent Duncan home, even though he had a fever and had traveled from West Africa. By the time he returned to the hospital, his symptoms had worsened. He died while being treated by medical staff, including the two women who have contracted the disease.
"I don't think we have a systematic institutional problem," Dr. Daniel Varga, chief clinical officer of Texas Health Resources, told reporters, facing questions about the hospital's actions.
Medical staff "may have done some things differently with the benefit of what we know today," he said, adding, "no one wants to get this right more than our hospital."
But on Wednesday night, the hospital released a statement offering a room to any employee who is concerned about exposure to Ebola.
"Texas Health Dallas is offering a room to any of our impacted employees who would like to stay here to avoid even the remote possibility of any potential exposure to family, friends and the broader public," the hospital said.
"We are doing this for our employees' peace of mind and comfort. This is not a medical recommendation. We will make available to our employees who treated Mr. Duncan a room in a separate part of the hospital throughout their monitoring period."
People in Vinson's office building were informed when officials went door to door, and also through early morning reverse 911 calls, officials said.
The health care worker had no pets, authorities said.
Official: Duncan should have been moved
An official close to the situation says that in hindsight, Duncan should have been transferred immediately to either Emory University Hospital or Nebraska Medical Center in Omaha.
Those hospitals are among only four in the country that have biocontainment units and have been preparing for years to treat a highly infectious disease such as Ebola.
"If we knew then what we know now about this hospital's ability to safely care for these patients, then we would have transferred him to Emory or Nebraska," the official told CNN senior medical correspondent Elizabeth Cohen.
"I think there are hospitals that are more than ready, but I think there are some that are not."
CNN's Rene Marsh, Tina Burnside, Miriam Falco, Caleb Hellerman, Devon Sayers and Chandler Friedman contributed to this report.