But the survey also suggested a drop in vaccine hesitancy, with 73 per cent of Australians in June agreeing or strongly agreeing that they would get a COVID-19 vaccine when it becomes available, an increase from 68 per cent in May.
Vaccine hesitancy is expected to have declined further due to Sydney's lockdown, which has now been extended to 30 July.
The main reasons people may not get a COVID-19 vaccination were concerns relating to potential side-effects (52 per cent) and effectiveness of the vaccine (15 per cent).
Meanwhile, Prime Minister Scott Morrison continues to blame Australia's expert medical advisors for the country's troubled vaccine rollout.
He also argues he did not want to rush approval of vaccines while case numbers were low.
"We have had a cautious approach in Australia on medical advice," he told ABC radio on Thursday.
"We wanted to follow all the usual steps and processes to ensure the vaccines were appropriately qualified before they were used in the community."
Mr Morrison said the initial plan was to rely on AstraZeneca vaccines that could be manufactured in Australia.
He continued to shift blame to the Australian Technical Advisory Group on Immunisation (ATAGI) for the slow rollout.
In April, ATAGI's updated recommendations saw the government make Pfizer the preferred shot for people aged under 50 because of AstraZeneca's link to rare blood clots.
In June, that recommendation was broadened to anyone aged under 60.
Pfizer remains the preferred vaccine for people younger than that.
But ATAGI updated its advice again this week because of the Sydney outbreak.
Under 60s in hotspot areas were advised to consider getting AstraZeneca if Pfizer was unavailable.
Australians living in areas impacted by coronavirus outbreaks have also been advised to bring forward their second dose of AstraZeneca.
"We received medical advice that has changed on two occasions and that medical advice, as I made very clear to ATAGI at the time, was based on an assumption cases would remain low," Mr Morrison said.
"The balance of risk assessment were based on low case numbers in Australia.
"It has created some confusion in the public."
Debra Petrys, the consumer representative on ATAGI for nine years before her retirement in June, told The Guardian Australia she was “disappointed” by the comments, describing them as “a bit unfair”.
“This isn’t the time for a blame game – everyone has made the decisions they’ve made, which were the best decisions with the evidence present at that time,” she said.
Ms Petrys defended ATAGI's decision to advise those under 50 to wait for the Pfizer vaccine.
“When we made the first decision in April was well before Delta came into Australia and, looking at the risks and benefits, I firmly believe it was the right decision," she said.
With reporting by SBS News