And NHS Borders has taken the “difficult decision” to cancel non-urgent operations for the rest of this week and next, under “very high demand” for beds.
Officials urged people to be cautious despite the move beyond level zero and warned the new freedoms may be “short-lived” if they are not.
It comes as Scotland reported 1,498 new cases on Wednesday – the highest daily total in almost three weeks – since July 23.
The number of people dying of Covid has also slightly increased, to 51 in the week to August 8, five more than the previous week, according to the National Records of Scotland.
NHS Highland saw more than 400 cases in the first week of August, said director of public health Dr Tim Allison, a doubling of the number reported in the previous week.
On the cancellation of non-urgent operations for NHS Borders, a spokesperson said: “We fully recognise the distress caused by this unavoidable decision and are truly sorry.
"We can assure you that the situation remains under constant review. All urgent, cancer and emergency surgeries will continue during this time.”
In the Highlands, Fort William is among the most affected areas, with 56 new cases since Friday.
The case rate per 100,000 people in Fort William South is 850, compared to 150 across Scotland as a whole.
NHS Highland said no single cause of the outbreak had been identified and it is not known whether the spike in cases has been caused by tourism.
Dr Allison said: “During the first week of August we had over 400 cases, which is quite a significant increase compared to previous weeks.
“It has been a very challenging few months for everyone and it is right that the new found freedoms we now have should be enjoyed. We can now catch up with family and friends, and travel a bit more.
"However, with relaxing of the restrictions and schools about to go back we are expecting a further rise in cases.”
NHS Fife has reported “considerable pressures” on healthcare services, due to Covid and non-Covid admissions.
Director of acute services Claire Dobson said July had been a “particularly challenging” month, with accident-and-emergency (A&E) visits more than a third higher than in January.
Despite this, most elective procedures have continued as planned, she said.
Other health boards have also reported high demand in recent days.
NHS Lanarkshire said it continues to see “sustained pressure” on services, while NHS Tayside said demand was still high.
Health secretary Humza Yousaf said last week that he expected case numbers to rise as a result of the move beyond level zero on Monday.
He said he could not guarantee there would be no further lockdowns in winter if Covid spiralled out of control.
The Scottish Government said it was too early to tell if the recent rise in Covid case numbers may be linked to reduced compliance with restrictions in the lead-up to the move beyond level zero on Monday.