Meanwhile, the mayor has warned the city’s residents that, amid Covid-19, the council does not "have the vaguest idea what our collective future will be".
He also suggested the Covid-19 pandemic could last 20 years and hit out at an academic who said councillors could vote down his choice as deputy.
Cr Rebecca Amundsen said the public’s perception of Sir Tim and his competence was changing.
"There is certainly a lot more critical assessment of the situation than it was before. Not just — I guess — a blind following of Tim."
However, while departing deputy mayor Toni Biddle shared concerns about Sir Tim’s performance, she believed he remained popular among the public.
"I think our voter turnout was 48%, which indicates half of our community doesn’t care about local politics," Cr Biddle said.
Those who did vote largely made their decision based on name recognition, which Sir Tim had in spades.
Both believed the best outcome for the city would be if Sir Tim resigned.
Cr Amundsen said she had made her position on Sir Tim clear to the independent evaluator, Richard Thomson, who was appointed amid a probe by the Department of Internal Affairs looking into "significant conflict" at the council.
"I think we are in a situation where a lot of us are speaking out about what is happening in council and I think [it] is a good thing."
In a newspaper column sent to the Allied Press-owned Southland Express, Sir Tim sounded a warning to residents.
"Without wanting to sound like the canary in the coal mine or a snake in the grass, I feel obliged to warn the Southland community that despite extensive workshops on our Long Term Plan, we don’t have the vaguest idea what our collective future will be."
He said the council was facing "the great unknown".
"We don’t know whether or not Covid-19 is the end of the great epidemic or only the beginning. We don’t even know whether it will last five years or 25 years."
He stressed Invercargill’s strengths, saying it was booming and had the best truck museum in the southern hemisphere.
He finished by saying: "And most of all — go the Stags."
Ms Biddle said she would not take any further action over the council’s problems.
Her last day at council is on November 15.
"I just want to be out," she laughed.
