The country is in a “very delicate situation”, Dr Eleanor Gaunt, research fellow at Edinburgh University’s Roslin Institute, said.
Some 202 people were in hospital and 20 were in intensive care.
Case rates remain highest in Dundee, with a seven day rate of more than 650 cases per 100,000 people.
Dundee is followed by East Lothian, Midlothian and Edinburgh, which has a rate of just under 550 per 100,000 people.
Dr Gaunt said: “We’re seeing right now an increase in cases – not just a linear increase in the number of cases, but they are really rapidly rising, and we don't know what the consequences of that are going to be as yet, so we're in a very delicate situation.”
“We should be proceeding with the highest level of caution.”
First Minister Nicola Sturgeon said the “steep rise” in cases showed people should be “really careful” and stressed the need not to be complacent.
But she added: "The positive news is that vaccination is providing a lot of protection, so we are not seeing a commensurate rise in the number of people going into hospital or intensive care, or dying from the virus."
Dr Gaunt instead stressed the rising case numbers and hospital admissions must be brought back under control with the rollout of the vaccination programme.
A total of 3.7 million people have now had a first dose of a vaccine in Scotland, with just under 2.7 million having had two doses.
“Right now we are not in a position of control, so we just need to press pause effectively and let the vaccination programme catch up with the number of cases and get the situation back under control,” Dr Gaunt said.
While the link has been weakened between cases and hospitalisations and deaths, Dr Gaunt pointed to long Covid as another serious consequences of high case numbers, which is not yet well understood.
Other neurological complications are also of “serious concern”, she said.
National clinical director Jason Leitch suggested sports stadiums may be allowed to operate as normal in August, but Dr Gaunt said it would be wiser to wait for the vaccination programme to “catch up”.
Some 60 per cent of adults in Scotland have been given both doses of vaccine.
It comes after the rollout dipped to its lowest weekday vaccination total since March on Wednesday.
Dr Gaunt added: “There's also the risk of new variants. If we put the virus into a high-pressure situation, where it's circulating in a partially-vaccinated population, then there's the risk that a variant emerges that's able to evade vaccine-induced immunity.”
The number of new cases of Covid-19 reported each day in the UK have now risen to the highest level since late January, new figures show.
A total of 22,868 cases were reported by the UK Government on Monday, the highest daily rise since January 30 when there were 23,275 cases reported.
The government also said a further three people had died.
The UK’s new health secretary Sajid Javid insisted the UK Government saw “no reason” for restrictions in England to go beyond July 19 – the same date the Scottish Government has targeted the country entering level zero.
The new health secretary told MPs on Monday the nation had to “learn to live with” the virus and suggested there would be no further extension of the restrictions.
The Scottish Government has been more cautious under the country’s own route map out of lockdown, aiming for all restrictions gone by August 9.
Making his first speech since replacing Matt Hancock, Mr Javid was upbeat about the restrictions finally ending.
He said: “While we decided not to bring forward step four, we see no reason to go beyond July 19, because, in truth, no date we choose comes with zero risk for Covid.
“We know we cannot simply eliminate it, we have to learn to live with it.
“And make no mistake, the restrictions on our freedom, they must come to an end.
“We owe it to the British people, who have sacrificed so much, to restore their freedoms as quickly as we possibly can and not to wait a moment longer than we need to.
“With the numbers heading in the right direction, all while we protect more and more people each day, July 19 remains our target date.
“The Prime Minister has called it out ‘terminus date’. For me, July 19 is not only the end of the line, but the start of an exciting new journey for our country."
But Jonathan Ashworth, Labour’s shadow health secretary, said the UK Government risked more people catching coronavirus and developing so-called “long Covid” if they lifted restrictions while case numbers were still high.
Responding to Mr Javid’s commitment on July 19, Mr Ashworth said: “A word to the wise – I’ve responded to a lot of these statements these past 15 months.
“I remember ministers telling us there was nothing in the data to suggest June 21 wouldn’t go ahead, I remember children returning to school for one day before the January lockdown, I remember ‘it will all be over by Christmas’. I remember ‘we will send it packing in 12 weeks’.
“Well, we’ve seen around 84,000 cases in the last week, an increase of around 61 per cent. Today we’ve seen the highest case rate since January.
“If these trends continue, we could hit 35,000 to 40,000 cases a day by July 19.”