Mr Gutwein said the three-day snap lockdown had "dislocated so many peoples' lives" and he was "annoyed and angry" after the COVID-positive man escaped hotel quarantine.

"We have a system of GTG passes, this person was denied entry to Tasmania and he still showed up," Mr Gutwein said during a COVID press conference on Saturday.

"I request that [Tasmania Police] consider what other action can be taken against this person and anyone else who has been an accomplice."

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Currently, the COVID-positive man has been hit with two fines totalling just over $3000.

One infringement is for breaching quarantine and the other is for attempting to enter the state without necessary approvals.

The person who aided his escape from the Travelodge Hotel Hobart has not been fined.

However, Mr Gutwein said the behaviour of those involved warranted harsher penalties.

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"I think their behavior's been outrageous," he said.

"I believe very strongly in light of the circumstances we should throw the book at the people who flout the rules."

Deputy Commissioner Donna Adams said Tasmania Police had seen high levels of compliance in the Tasmanian community, but the COVID-positive man had been "uncooperative and difficult to deal with".

Deputy Commissioner Adams said after the conclusion of the public health response, police would continue to investigate the man for any other offences.

"We will investigate this man's actions and any other examples of offending will be investigated," she said.

She said police had the ability to bring the man before a magistrate, where he could face a maximum fine of $17,000 or up to six months' imprisonment.

Southern Tasmania's lockdown began at 6pm on Friday and police confirmed they had set up a number of checkpoints at main arterial roads.

In the last 16 hours, police confirmed that 28 people had been provided with advice, and six people were issued infringement notices after being provided advice and failing to comply.

At police checkpoints, 45 vehicles had been turned around for failing to meet the requirements to leave the area.

Tasmania Police Assistant Commissioner Jonathan Higgins reiterated police would be highly visible in Southern Tasmania for the three days.

"We know the majority of people will do the right thing because they understand how important these measures are for protecting our community," he said.

"To those people who are considering flouting the rules - we are using significant police resources to patrol southern Tasmania to make sure people are adhering to the restrictions.

"Police will be taking a strong approach in dealing with those in the minority who continue to flout the rules and put other Tasmanians at risk."

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