Dr Sutton said there was an obligation across the entire state for people to minimise their interactions with each other.
"Though the stay at home directions apply to those restricted postcodes, all of us across Victoria have to really consider whether we need to see people in other settings, other households, including family members and friends," he told reporters on Thursday morning.
Stay-at-home orders have been reintroduced for 10 postcodes in Melbourne's inner north and west from Thursday until at least July 29, after a 15th consecutive day of double-digit case increases in the state.
People living in those neighbourhoods will only be allowed to leave home for work, study, essential shopping, exercise or to receive or give care.
Police will be patrolling the streets and screening residents entering and leaving the postcodes.
Victoria's chief health officer Brett Sutton 'really hopes' further suburbs wouldn't be forced into lockdown
Victoria finds student-to-student transmission
Dr Sutton said there was evidence to indicate the virus was being spread between school students.
"There has been some student-to-student transmission and also teacher-to-teacher transmission at some schools," he said on Thursday.
"When there's a heavy load of community transmission - more students getting infected - those schools have closed."
Children were less likely to spread the infection and remained a lower priority for testing, Dr Sutton said.
Federal Health Minister Greg Hunt said there had been "clear failures" in Victoria leading up to the latest outbreak.
"Right now the offer is there if additional support is required. It's a matter for them, how they accept that, he told the ABC on Thursday morning.
Questions raised over hotel breaches
New revelations came to light on Thursday regarding cases of the virus linked to quarantine breaches in Melbourne hotels.
An inquiry into the state's hotel quarantine program, headed by former judge Jennifer Coate, is under way after it was revealed the program fuelled outbreaks in the city's inner north and west.
Companies charged taxpayers for shifts never worked, leading to fewer security workers on duty and a higher risk of infections, The Herald Sun reports.
There are also claims some guards were sleeping with guests, shook hands and shared lifts, and allowed guests to move between rooms.
A security guard who worked at one of the designated quarantine hotels told the Nine Network he only received five minutes of training before taking up the role and some guards had worked at other hotels, potentially spreading the virus further.
Should Australians be wearing masks?
Prime Minister Scott Morrison said he was concerned about the outbreak but backed the drastic lockdown measures adopted by the Victorian government.
"That was the appropriate response and they have our full support in implementing it," he told reporters on Thursday.
"I think it's important to learn the lessons and I think Premier Dan Andrews has been, I think, quite up-front about that.
"So he has my support to continue to put these measures in place and get on top of this outbreak and we've got to focus on the problem."
Meanwhile, NSW has announced Victorians from the 10 postcodes could face six months in jail or fines of up to $11,000 if they try to cross the border.
"Victorians right now from those hot spots are not welcome in NSW. We are sorry," NSW Health Minister Brad Hazzard said.
Queensland is banning all Victorians from entering the state but welcoming other visitors from July 10.
South Australia has shelved plans to reopen its Victorian border but is weighing up a travel deal with NSW and the ACT.
There were 87 new cases across the country on Wednesday, taking the total to 7920.
After an adjustment in reporting methods, there are fewer than 400 active cases in Australia, down from 693 on Tuesday.
More than 350 active cases are in Victoria.
People in Australia must stay at least 1.5 metres away from others. Check your state’s restrictions on gathering limits.
Testing for coronavirus is now widely available across Australia. If you are experiencing cold or flu symptoms, arrange a test by calling your doctor or contact the Coronavirus Health Information Hotline on 1800 020 080.
The federal government's coronavirus tracing app COVIDSafe is available for download from your phone's app store.
SBS is committed to informing Australia’s diverse communities about the latest COVID-19 developments. News and information is available in 63 languages at sbs.com.au/coronavirus.