The new infections have raised alarm amid a slow take-up of vaccinations in Australia and as other countries in the region, including Singapore, Taiwan and Thailand, have seen cases increase into the hundreds or thousands in the past few weeks.
The family cluster - originally involving a man in his 30s, a man in 70s, a woman in 70s and a pre-school aged child from three households in the Whittlesea area - emerged on Monday.
Acting Premier James Merlino said the new case was a man in his 60s linked to the City of Whittlesea infection, which now totals five infections.
Authorities, however, said they could not yet find how the latest cases contracted the virus from the overseas traveller.
New restrictions have been imposed in the state, including the mandatory use of masks indoors and a five-person gathering limit at homes each day.
The health department last week admitted it had listed the wrong supermarket as an exposure site when the Wollert man tested positive earlier this month. Contact tracing of the new cases is underway.
Four previously reported cases have now recovered.
ACT Health has warned Canberrans to remain vigilant for COVID-19 and get tested even if they have mild symptoms. However, the exact train of transmission between the May 11 case and the new cases is not yet clear.
"An epidemiological link has yet to be determined between these cases and there is now no known link between people in the current outbreak and any of the exposure sites from Wollert".
The statement from Hipkins on Tuesday said that while the new case announced in Melbourne was not unexpected, New Zealand officials had assessed the situation and the most cautious option was to pause the travel bubble with Victoria "as there are still several unknowns with the outbreak".
Under new restrictions face masks must be worn indoors unless there's an exemption in place.
"Hopefully, they have a good sense of whose been exposed, what the risks are, so they can identify those people they need to call into quarantine, rather than go into something more wide-spread", she said.
Anyone who visited those locations during those times have been ordered to get tested immediately and isolate for 14 days regardless of the result.
Anyone in New Zealand who has been at any of the locations of interest at the specified times should contact Healthline on 0800 358 5453 for advice on self-isolation and testing as soon as possible.
Anyone who has visited Melbourne since 11 May should monitor for symptoms and seek advice from Healthline if any symptoms develop, Hipkins said.
Last night, NZ's Ministry of Health advised Kiwis who have recently travelled to Victoria to check a growing list of locations of interest.