Nine of the cases are being managed at home, while one person is staying at a community management facility.

The three new cases, which were confirmed on Monday morning, include a man in his 30s who arrived in Hobart via Melbourne on Wednesday.

The other two are a male overseas arrival in his 40s who touched down in Hobart on Friday via Melbourne and a man in his 20s who arrived in the Tasmanian capital from Sydney on Friday.

Several exposure sites in the state's south were declared on Monday.

People who were at the Groundsman Espresso cafe in Hobart from 8.09am to 8.19am on Friday have been declared casual contacts and told to get a test.

Hobart City Mission in the suburb of Moonah has also been declared a casual-contact exposure site on Friday from 11am to 1pm.

Five businesses in Sorell are considered low risk sites, with authorities urging people to monitor for symptoms if they visited at certain times on Friday and Saturday.

Tasmania had no virus cases when it reopened on December 15 to fully vaccinated travellers from hotspot jurisdictions.

The state government will be enforcing a mask mandate across all indoor settings, public transport and ride shares from 12.01am on Tuesday.

All people travelling from high-risk locations must return a negative test 72 hours before they arrive in Tasmania.

Several of the earlier cases in Tasmania have been linked to a superspreader event at a nightclub in Newcastle.

More than 91 per cent of Tasmanians aged over 16 are fully vaccinated.

Australian Associated Press