Data from the Office for National Statistics (ONS) shows that in the week ending June 17, an estimated 250,700 people in private households had the virus – equating to around one in 20.
This is up from around one in 30 people the previous week and is the highest level in the UK.
Covid levels are continuing to rise in all four nations of the UK, with the increase likely caused by infections of the Omicron variants BA.4 and BA.5.
Ms Sturgeon said she did not want to reimpose restrictions.
"We need to all of us be sensible and take precautions," she told the BBC.
"It's no longer, for example, a requirement to wear a face covering in an indoor place, but right now when cases are rising again, it would be a sensible precaution to wear a face covering if you're in an indoor place with crowds of people around you.”
Health secretary Humza Yousaf said rising infection levels were “of concern”, but he highlighted that Scotland had a highly vaccinated population.
“Our spring booster campaign has been immensely successful with 91 per cent of those over 75 years old and 86 per cent of elderly care home residents receiving their Covid vaccine,” he posted on Twitter.
“We can all play our part in keeping ourselves and others safe. Whatever dose you’re eligible for, it’s not too late to get vaccinated. Wear a face covering in indoor spaces and on public transport, keep indoor spaces well ventilated, practice good hygiene measures.”
It is estimated around one in 45 people in Wales had the virus in the week to June 18, one in 40 in England, and one in 30 in Northern Ireland.
Kara Steel, senior statistician for the ONS Covid-19 Infection Survey, said: “Rates have continued to rise across the UK, with the largest increase seen in Scotland.
“In England, infections increased across all age groups, with the lowest level of infection seen in school-aged children.
“These increases are largely driven by the Omicron BA.4 and BA.5 variants. We will continue to closely monitor the data.”
NHS Lothian said its hospital system was under “extreme pressure”.
Tracey Gillies, medical director of NHS Lothian, said: “The number of hospital inpatients who have tested positive has increased by 50 per cent since a week ago. This presents many logistical issues as they need to be cared for within Covid-19 specific areas.
“On top of that, and in line with community transmission levels, there are increased numbers of staff testing positive with Covid-19, who must then self-isolate to protect patients.
“All of this means services are stretched right across the system.”
ONS figures for Scotland are given for the week ending June 17, while for England, Wales and Northern Ireland they are for the week ending June 18.
Separate Public Health Scotland (PHS) data published on Wednesday showed cases of Covid-19 rose by almost a third in Scotland last week.