The latest case is a teenager who flew to Launceston from NSW on Thursday.
He was tested after NSW Health notified him he had been at an exposure site, but before the notification arrived he visited a cafe and a JB Hi-Fi store on Thursday afternoon.
Those two venues are now exposure sites, along with Launceston Airport baggage carousel.
Close contacts on board the man's flight and people who checked in to the venues he visited around the same time have been contacted by Tasmanian public health services.
Tasmanian Premier Peter Gutwein says initial investigations have identified 19 close contacts, and advised anyone in Launceston experiencing symptoms to get tested.
He says that he "would expect" the teen had received a negative PCR test before coming into Tasmania, as the entry conditions require, but did "not have that public health advice" when asked on Saturday.
Tasmania requires people arriving from high-risk areas to return a negative test in the 72 hours before they travel.
Two other confirmed cases are linked to an early Wednesday morning Qantas flight from Sydney, one of the first planes to land after the border reopened.
Another person who was confirmed positive on Friday flew from Melbourne on Wednesday afternoon.
State Public Health director Mark Veitch said two new cases on Friday had very little interaction with the community and went to get tested soon after being told they had been at an exposure site.
Recent arrivals from Newcastle are being contacted after the first three cases were linked to superspreader events in the town.
Positive wastewater samples have been collected from Norwood in Launceston, where there are no known cases.
Anyone in the area is advised to get tested immediately if they experience any symptoms, no matter how mild.
Mr Gutwein said on Friday the state government was considering a mask mandate at indoor venues and would have more to say in coming days.
Tasmania had no active cases when the state reopened.
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