Activist Julie Hepburn said there was “no doubt” Ms Sturgeon had a mandate for another referendum.
She said that mandate should be acted on before the next Holyrood election in 2021.
Inverclyde councillor Chris McEleny had previously been the only hopeful in the depute leadership contest calling for a swift referendum. She has demanded another vote by the end of next year.
Asked about the timescale for a potential second referendum on STV’s Scotland Tonight programme, Ms Hepburn said: “At the earliest possible opportunity. There’s no doubt that we have a mandate to have a referendum.”
Quizzed on whether that decision should be taken during the existing Parliament, she added: “Yes, of course, of course.”
Ms Hepburn continued: “The case for independence has never been stronger. We fought the last referendum about why Scotland should be independent.
“The next time it’s going to be about why we need to be independent as a matter of urgency.
“The First Minister will make that decision based on a number of factors, but she is more likely to have an earlier referendum if we are out there actually campaigning for independence now.
“So I don’t think we need to wait for a date. I think we need to get out there, launch a fresh campaign for independence.”
Ms Sturgeon is due to update MSPs on a “precise time scale” in the autumn.
The First Minister initially announced plans for a referendum in March last year.
She was forced to “reset” her plans after the SNP lost 21 MPs in the snap election in June last year.
Economy secretary Keith Brown, considered the frontrunner for the SNP deputy role, has said he wants to “wait and see” what emerges from the Brexit process before declaring his support or otherwise for a second Scottish independence referendum.
Mr McEleny has conversely said the key role of depute leader “on day one” should be “getting the new Yes Scotland campaign up and running”.
“Just look at what’s happening to the UK at the moment within a Scottish context,” he said.
“Our parliament and devolution is under threat. Child poverty is increasing, food bank use is on the increase and that is all on the watch of the UK Government.
“I don’t think there’s a better time than right now to have an independence campaign. We’re ready to go.
“That’s why we should have that referendum and I think people across Scotland are ready to grasp their own future with two hands this time.”