Half of the new cases arrived from NSW, including one transiting from overseas.
Another three are from Victoria, including someone from overseas, and one more arrived from Queensland.
One case is a close contact of a previously confirmed infection and investigations are underway to find the source of the final case.
The infections, reported on Wednesday, bring the island state's number of active cases to 26.
All have been recorded since the island reopened to fully vaccinated travellers from the mainland on December 15.
Eighteen cases are being managed at home, four are in a community-case management facility and four are still being assessed or processed. None are in hospital.
The state government opened additional mobile testing clinics in Hobart and Launceston on Wednesday to deal with a spike in demand.
Premier Peter Gutwein has defended the state government's reopening plan, after the Greens called for Tasmania to again close to arrivals from Victoria, NSW and the ACT.
"I can understand that there is some anxiousness and that's quite OK," he told reporters on Tuesday.
"As we move forward we need to learn to live with COVID. While some people would prefer us to stand still, we need to keep moving forward."
National cabinet on Wednesday did not make stronger indoor mask mandates or shorten the time for COVID-19 vaccine booster shots.
Tasmania on Tuesday implemented a mask mandate for all indoor venues, plus public transport and ride-sharing services.
Following national cabinet's meeting on Wednesday afternoon, Mr Gutwein said he urged NSW Premier Dominic Perrottet to reinstate some public health measures after the state recorded 3763 new cases.
"New South Wales seems to be the one outlier in all of this," Mr Gutwein told reporters.
"I don't think that it's appropriate that no contact tracing other than in high-risk areas occurs and they haven't as yet mandated masks."
He also encouraged Tasmanians to book in for their booster shots, with only 10 per cent of people so far getting the third dose.
More than 90 per cent of Tasmanians aged over 12 are fully vaccinated.
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