New information has come to light relating to investigations across 13 institutions as well as "reference to other hospitals", the Health Secretary said.
He has asked police to review all of the evidence before relevant information is passed on to investigators "as quickly as possible".
Investigations were launched into activities at Broadmoor, Stoke Mandeville and Leeds General Infirmary following the abuse revelations last year, along with inquiries at 10 other trusts.
But further "relevant information" regarding the presenter's activities has emerged, the Department of Health said.
Mr Hunt said any hospitals involved would be named once the review of the new potential evidence had been completed.
"We understand the material includes information about hospitals where investigations are already under way and reference to other hospitals," he said in a statement to Parliament
"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 added: "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, who died in 2011 aged 84, is believed to have abused hundreds of children. He had a bedroom at Stoke Mandeville Hospital, an office and living quarters at Broadmoor, and widespread access to Leeds General Infirmary.
The wave of allegations made against him sparked the launch of the Operation Yewtree police investigation, which is also looking into claims about others linked to him as well as separate allegations about a number of high-profile figures.
Lesley McLean, divisional manager for Victim Support in West Yorkshire, said: "Victims have shown a great deal of courage in coming forward to report dreadful experiences from many years ago. Victim Support staff and volunteers in West Yorkshire have been working closely with some of those affected by Savile's criminal activities, and have heard distressing and disturbing accounts of what happened to them.
"Reliving past abuses can be extremely traumatic for victims, who have had to cope with these events for many years. While getting to the truth of what happened is important, we continue to make sure that victims come first, and are given the emotional support they need as this investigation progresses."
Liz Dux, head of abuse at Slater and Gordon Lawyers, who represents 72 potential Savile victims, said: "While the victims are encouraged by news that the NHS inquiries will be comprehensive investigations of other institutions that may have been involved, the news that the time-frame for concluding the investigation has been extended until next June will be met with dismay.
"My clients all need closure and whilst we welcome a detailed understanding of how Savile was able to operate unchecked for so many years, at the same time we need to recognise that until these reports are concluded their suffering continues."