Black Hills Power said nearly 12,000 people were still in the dark, including about 6,500 customers in the Rapid City area. The total was down from 28,000 customers at the peak of the outage, but some homes were not expected to get the electricity turned back on until the end of the week.
"We've made some substantial progress over the last two or three days," Vance Crocker, vice president of operations for the utility, told the Rapid City Journal.
Black Hills Power said it had more than 200 contractors working to remove downed branches from power lines.
Hundreds of other homes served by electric cooperatives also were without service, the Journal said.
The storm that began Friday and lasted into the weekend also forced the cancellation of classes at area schools, a situation that was to continue Tuesday in Rapid City.
South Dakota's National Guard was put to work helping restore power, using heavy equipment to drag power company trucks to where they need to go, the newspaper said.
Rapid City Fire Chief Mike Maltaverne urged city officials Monday to declare a state of emergency, which could lead to federal aid. The city got up to 30 inches of snow.
"In my 24 years in Rapid City this storm, in my recollection, is unprecedented, certainly on behalf of the Fire Department," Maltaverne told the council at an emergency meeting. "We reached a level Friday and into Friday night that I certainly haven't seen in my tenure at the department in terms of the scope of the storm, the lack of access that we had."
He said "dozens upon dozens" of calls for service went unanswered.
"We literally had to leave our equipment out in the field," Maltaverne said. "That was not a comfortable position to be in as a fire chief. I had assets out in the field that we can't use and literally the rescuers were in need of being rescued. The city was in bad shape for a little while."
City Street Superintendent Don Brumbaugh said about 80 percent of the city streets had been plowed, with the remainder expected to be cleared Monday night.