Circular Head mayor Daryl Quilliam said it was "absurd" the government would take public health resources away from their post for the transport rather than just providing testing in Smithton.

Health Minister Jeremy Rockliff said the government was exploring "all feasible options... to balance resourcing".

The woman, who asked not to be named, learned on Monday she was a close contact of the confirmed COVID case who had visited the Smithton cafe Hugo's Brew and Chew.

She took it upon herself to contact Public Health, and informed them she was her three-year-old daughter's sole carer and did not have her own transport option to get to Burnie to be tested.

On Tuesday at 8am she was contacted by a St John Ambulance representative, who told her they would be providing her transport to the testing site, but by 1pm she had not been collected.

The Burnie testing clinic, now run by private medical provider North-West Pathology since December 13, is only open between 8am and 12pm until January 3.

The woman said she was contacted by an Ambulance Tasmania representative on Tuesday afternoon, who said they would facilitate her transport on Wednesday morning.

"It seems like a whole lot of effort to arrange transport just for us, and presumably other people who don't have transport," she said.

"It's a two-hour round trip to Burnie."

She said she was quarantining with her daughter at home, and they both had cold and flu-like symptoms, but she could not yet be certain her daughter had not just picked up a cold at daycare.

She was hopeful testing opportunities and facilities would be improved before COVID in Tasmania got "out of hand".

Cr Quilliam had called on the government to provide short term mobile testing in Circular Head following the listing of exposure sites on Sunday.

"I don't think we need to have testing there for weeks on end," he said on Tuesday.

"But I feel they should come down for a couple of days at least. Give people a chance to get tested."

He said diverting Ambulance Tasmania resources away from Circular Head could also be problematic if, for example, there was a serious car crash in the region.

"I think it's absurd. They ought to make testing available in Smithton."

Hugo's Brew and Chew Owner Adele Hugo said she had purchased $100 worth of rapid antigen tests for those around her and directed her staff to be tested in Burnie.

All the tests had returned negative results.

Despite that, she and her husband had made the decision to not reopen the cafe until the new year.

Mrs Hugo said she believed she could have been made aware sooner that the cafe was an exposure site.

She said she would not have chosen to spend Christmas with vulnerable people and a healthcare worker had she known.

Mr Rockliff was asked if using Ambulance Tasmania resources to transport individuals two hours for testing was sustainable, but said it was "not appropriate to comment on individual cases".

However, he did say testing capacity had increased across Tasmania since Tuesday December 21, but did not commit to providing mobile testing to Circular Head.

"The Department of Health continuously monitors the postcodes of calls to the Public Health Hotline and bookings on their website to gauge where there is a need for our mobile testing clinics," he said.

"[We] will continue to closely monitor our public health response, and explore all feasible options, including additional partnerships, in order to most appropriately balance resourcing where required."