The proportion of reinfections in that month was 9.8 per cent, a huge increase from the 0.6 per cent recorded through the rest of the pandemic before the arrival of the Omicron variant.
It comes as a separate report showed the proportion of patients admitted to intensive care units (ICUs) who were under 40 tripled in the third wave compared to the first and second, and that unvaccinated people were six times more likely to be admitted to ICUs with Covid.
Experts believe the Omicron variant is less susceptible than previous strains to natural immunity brought on by having the virus.
Public Health Scotland has begun to publish data for the first time in a bid to monitor these reinfections, defined as a case of the virus at least 90 days after previously testing positive.
Throughout the pandemic, some 34,774 cases of reinfection have been recorded.
The proportion of reinfections remained low in the first and second waves, with small peaks of 3 per cent in July 2020 and 2 per cent in April last year.
But this percentage rose to 6.4 in December 2021, and again to 9.8 last month.
The highest number of reinfections have been reported in younger people, aged 20 to 24, in line with overall case rates.
While the change in England has already come into effect, this will be echoed by Public Health Scotland later in February, meaning daily case rates will not be comparable between the nations until then, Ms Sturgeon said.
"This data on reinfections obviously becomes more important as the pandemic progresses and more people get Covid for a second time,” she said.
“However, it is important to stress that, as things stand, the current daily figures do capture the significant majority of people who test positive each day.”
A new report from the Scottish Intensive Care Society Audit Group showed 1,087 patients have been admitted to intensive care in the third wave of Covid, defined as from May 18.
One in five of these patients (22 per cent) were under 40 years old, compared to 8 per cent in waves one and two.
Mortality at 30 days after admission was slightly lower, at 30 per cent compared to 37 per cent.
Unvaccinated people were around six times more likely to be admitted to ICUs with a positive Covid test than people who had received two vaccine doses.
Meanwhile, a survey of ICUs on January 13 found 28 per cent were unable to maintain recommended nurse-to-patient staffing ratios with ICU-trained registered nurses from their own unit.
Just 73 per cent of the expected number of ICU-trained registered nurses from their own unit were working on the unit on the day of the survey, which was undertaken when health boards across the country were struggling to cope with high levels of staffing shortages.