The capitulation came after doctors threatened to boycott My Health Record to protect patients' interests, by refusing to upload data.
The minister has also promised to amend the My Health Record Act 2012, to ensure that no patient's medical records may be accessed by any third party without a court order.
As currently worded, the legislation allows authorities including police, courts and the Australian Taxation office to access patient data without a warrant. This will be changed.
Mr Hunt also pledged to ensure that any patient who wished to delete their My Health Record could do so permanently.
The concessions - which have been dismissed by former AMA president Dr Kerryn Phelps as "woefuly inadequate" - followed lobbying by the Australian Medical Association and the Royal College of General Practitioners, which have welcomed the changes.
But many doctors remain concerned about remaining provisions of the legislation - including Section 98, which gives the "system operator", currently the Australian Digital Health Agency, the power to delegate "any function" to “any other person” with consent of the minister.
"This could include the disclosure of your health information," Dr Phelps said.
Legal experts have backed the unions' claim about potential employer and insurer access, saying the laws governing access were spread across several pieces of legislation and the rules were unclear.
An online petition calling on the minister to switch to an opt-in My Health Record system, as originally planned by the then Gillard government in 2012, now has more than 50,000 signatures.
Mr Hunt said the opt-out period extension would give Australians more time to consider their options as the government "strengthened" the legislation to match departmental policy.
"The Australian Digital Health Agency’s policy is clear and categorical - no documents have beenreleased in more than six years and no documents will be released without a court order. This willbe enshrined in legislation," he said.
"Healthcare providers cannot be authorised to collect, use or disclose a healthcare identifier, and as a consequence access a patient’s My Health Record, for employment and insurance purposes.
"Under the Act it is expressly prohibited and using or disclosing a healthcare identifier withoutauthority is an offence and subject to severe penalties, including two years in jail and a fine of$126,000."
The federal opposition has called for a suspension of the system while concerns are addressed.
However, Mr Hunt last week secured the support of his state counterparts, including the Queensland Labor government, during a COAG meeting in Alice Springs.
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