Mr Hunt said that new information had come to light relating to investigations across 13 institutions as well as "reference to other hospitals".
He said that he had asked police to review the new evidence "as quickly as possible" so it could be passed on to investigators, but that it would lead to a delay in the completion of the reports.
The health secretary is overseeing three separate inquiries into Savile's sex offences at Stoke Mandeville, Leeds General and Broadmoor hospitals, along with inquiries at 10 other trusts.
Savile carried out fund-raising events to raise millions for hospitals
However, Mr Hunt said that new "relevant information" regarding the presenter's sexual assaults had emerged, which included allegations about hospitals not already under investigation.
He said that any hospitals involved would be named after the review of the new evidence.
In a statement to parliament, Mr Hunt said: "We understand the material includes information about hospitals where investigations are already under way and reference to other hospitals.
"Once this review is complete, the information will be passed on to the relevant trusts or investigations as quickly as possible."
The final reports of the current investigations are now expected to be finished by next June.
He said: "Although all 13 investigations are currently on track, this additional material means that the timetable will be affected. It is vital that the final NHS investigation reports are thorough and complete and reflect all the evidence about Jimmy Savile's pattern of offending."
Savile's sexual offences spanned 1955-2009
In January, a joint report by the Metropolitan Police and the NSPCC found that 450 people had made allegations against the late DJ, who died in October 2011 at the age of 84.
His crimes spanned more than 50 years, and were carried out across 28 police forces on victims aged between eight and 47.
Many of his assaults were against vulnerable patients at the hospitals where he carried out volunteer work and enjoyed unprecedented access.
According to the NSPCC and Met Police report, 22 offences were carried out at Stoke Mandeville, one at Broadmoor and 16 at Leeds General Infirmary. Savile carried out a total of a further 10 offences at 10 other NHS institutions.
Savile had his own room at both Stoke Mandeville Hospital, in Buckinghamshire, for which he had raised millions of pounds through his charity fund-raising events, and at Broadmoor hospital. This despite concerns being raised at both institutions about his behaviour.
Caroline Moore has told how she was attacked by Savile at Stoke Mandeville
Caroline Moore, now in her 50s, was 13 years old when she was sexually assaulted by Savile at Stoke Mandeville Hospital. Paralysed and in a wheelchair, she was sitting in a corridor when he struck.
"He came over and didn't speak to me, he just bent down and rammed his tongue down my throat.
"After he'd done it he just got up and walked away. It was all very matter-of-fact.
"I was horrified. I didn't really know what had happened to me but it's stayed with me all these years."
