Hammond was driving a jet-powered car, capable of reaching 595 kilometres per hour, when it crashed during a shoot for the programme at the disused Elvington airfield near York.
The 36-year-old is in a serious but stable condition at Leeds General Infirmary, where friends and family are keeping a bedside vigil following Wednesday's accident.
Doctors said Hammond's injury gave them "cause for concern" but added they were "reasonably optimistic" he would make a good recovery.
Hammond was airlifted to hospital after the record-breaking Vampire dragster veered off the runway, flipped over several times and crashed on to grass.
It's thought he was travelling at speeds close to 482.8 kilometres an hour.
The BBC is reporting that police and the Health and Safety Executive (HSE) are investigating the incident and the BBC itself has begun an inquiry.
Event organisers insisted that proper safety precautions were taken.
Barb provokes smile
A man believed to be Hammond's brother, spoke to reporters at the hospital in Leeds, and said he was upbeat about the father of two's progress.
He said: “He's making a bit of good progress. Both his brothers are here and his wife. The doctors are very pleased.”
Hammond's wife Mindy was at his bedside in the Leeds hospital and he was also visited by Top Gear co-hosts Jeremy Clarkson and James May.
Clarkson told The Sun newspaper that Hammond had smiled when Clarkson had insulted his driving skills.
“I said: ‘The reason you’re here is because you’re a c**p driver’,” Clarkson was quoted by The Sun as having said.
“He then smiled at me. It was an amazing moment, very moving,” Clarkson told the newspaper.
Clarkson also said the presenter had no visible injuries following the ordeal he had been through other than a black eye.
The presenter had been driving a jet-powered dragster similar to the Vampire - used by Colin Fallows to set the British land speed record of 482.8 kilometres per hour.
Hammond had to be cut free from the car after eyewitnesses at the Elvington airfield near York described how he "veered off to the right" and the car's parachutes were deployed.
Primetime Land Speed Engineering, which is jointly run by Fallows, organised the event.
A spokesman for the firm said the vehicle Hammond was driving on Wednesday "had been prepared and was being operated to the highest of standards".