The driver, 65-year-old Harry Caligone, was the only person killed in the crash, and a medical examiner's report could reveal whether he had a health problem that could be a factor in the crash, said Capt. John Malone of the Alabama Law Enforcement Agency.
About three dozen people went to hospitals after the accident, and nine remained hospitalized Wednesday.
The bus, carrying 46 students and adults from Disney World to Channelview High School in suburban Houston, unexpectedly left the westbound lane of Interstate 10 before dawn Tuesday.
Malone said the bus veered across the median into the eastbound lane before shooting back into the grass median and driving off the ravine.
"It actually hit the guard rail on the eastbound side and was directed back into the median," he said.
There were no signs of skid marks, Malone said, indicating the driver didn't apply the brakes in a hurry.
"It just rolled," Malone said.
The National Transportation Safety Board said the bus was new and making only its second trip.
Pete Kotowski, an investigator with the agency, told a news conference that Caligone was on his way to meet a relief driver less than an hour away in Mobile, Alabama, at the time of the crash. Switching drivers is a normal procedure, he said, and the driver had not exceeded the total number of hours he was allowed to drive.
The crash occurred on a rural stretch between Pensacola, Florida, and Mobile, Alabama.
A student told the media that the band director called the driver called out "Harry" several times before the crash, but Malone said investigators haven't determined what happened aboard the bus.
Caligone's sister-in-law, Angela Caligone, 58, of Houston, said Caligone had been a bus driver for more than 20 years, the last 15 years with First Class Tours Inc. of Houston.
Caligone said her brother-in-law had just passed a physical with "flying colors." A company spokesman said Caligone had a checkup earlier this month but didn't provide details of the results.
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