The island state reported 867 fresh infections on Wednesday, a new daily record, taking the number of active cases to 3118.
State Public Health Director Mark Veitch said about one in 170 Tasmanians is an active case.
"I would not at all be surprised if in a week's time as many as one in 50 Tasmanians are on the list as a current active case of COVID," he told reporters.
Tasmania had no cases when it reopened to high-risk mainland jurisdictions on December 15.
Dr Veitch said case numbers had been on a "steep increase" over the past week and had doubled roughly every two or three days.
"It's difficult to estimate what the true numbers of COVID are at the moment. It could be as many as twice as many people as we're actually diagnosing," he said.
Five people with coronavirus are in hospital for unrelated medical conditions, while 333 cases are being managed at home and 72 are in a community facility.
"Our per capita rates of COVID are similar to Queensland, SA and the ACT, a bit less than Victoria and quite a bit less than where NSW is at the moment," Dr Veitch said.
"We're expecting this wave to continue rising for a week or two at least yet. Unfortunately we are likely to see some hospitalisations in the coming weeks."
Dr Veitch flagged a change to how positive cases are defined to find a "feasible, sustainable and largely automated approach to managing mostly mild cases and their contacts".
"Cases are currently being defined as having a positive PCR test," he said.
"It is possible that in the coming weeks we will look to diagnoses based on rapid antigen tests or perhaps in some instances on clinical symptoms. That's already being considered in other parts of Australia."
Fifty staff from the state's major hospital, the Royal Hobart Hospital, have tested positive.
Department of Health Secretary Kathrine Morgan-Wicks indicated some specialist services at the facility had been impacted.
"We have not had any reports of hospital transmission of COVID. This has been staff that have experienced community transmission or have been on leave," she said.
Tasmania's previous daily high was 702 cases, reported on Tuesday.
"This is challenging. I expect by now most Tasmanians will know someone with COVID or someone who is a close contact," Premier Peter Gutwein said.
Dr Veitch said public exposure sites would no longer be listed on the state government's coronavirus website.
"It's absolutely important that the public understands wherever you go in Tasmania over the coming weeks ... there's a very good chance you're been somewhere where there is someone with COVID."
The state government announced it was increasing capacity at the Macquarie Point drive-through site in Hobart due to high demand for testing.
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