The storm that swept the coast of Massachusetts Saturday night brought only 4 inches of snow to Boston, but more than a foot to some towns south of the city and left thousands without power.

Crews have been working to get power systems back on track, but mid-afternoon, about 4,859 customers remained off the grid, with those hardest hit on the mid-Cape.

And there’s more snow to come on Tuesday.

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“It’s another storm that will move up from the south,” said Charlie Foley, meteorologist with the National Weather Service in Taunton, “but it won’t be the magnitude of what we had Saturday night.”

The snow is expected to begin falling late Tuesday morning or early afternoon. One to 3 inches are expected in Boston, and Foley said that’s a worst-case scenario.

After that, there will be a temporary reprieve from the cold: Wednesday, Thursday, and Friday are expected to warm up significantly, with the high temperatures on Friday potentially reaching the 50s.

The warmer temperatures will be good news for residents hit hard by the barrage of snowstorms. Especially on the Cape, the most recent accumulation may have been the most harmful.

“This storm, of all the storms we’ve had this winter, has caused the most damage in terms tree-related power outages,” said NStar spokesman Mike Durand.

This storm, which hit Cape Cod the hardest, blanketed Sandwich with more than 15 inches of wet compact snow, Foley said.

Crews have been working through the night and restored power to more than 30,000 customers, but about 5,000 are still waiting, including 689 in Brewster, 650 in Yarmouth, 562 in Barnstable, 510 in Dennis, and 588 in Orleans.

Western Massachusetts Electric Company and Public Service of New Hampshire are assisting on Cape Cod.

“Winds are still strong on the Cape, but given the resources we should have power restored to most costumers by tonight,” Durand said.

Only a handful of National Grid customers lost power Saturday night, and all have been restored.

“Based on our service area and the track of the storm, while we prepared for the worst, we certainly made out the best,” said National Grid spokeswoman Debbie Drew.

Dispatchers at Sandwich police and fire departments said the town had a few minor car accidents and brief power outages, but no lasting damage and nothing more significant than what’s normal in any winter storm.

Sergeant Walter Warren of the Yarmouth Police Department said the storm’s wrath had the biggest effect on the town’s electricity infrastructure.

“The only damage we had was a lot of transformers and downed lines and downed trees and that’s about it,” he said this morning in a telephone interview.

Warren said about 100 power were lines down, a half-dozen transformers had been damaged.

“We’re very, very happy with the NStar response. They’ve gone above and beyond,” he said.

Boston fared well with few snow-related incidents reported.

“We really didn’t have any issues at all,” said Kate Norton, spokeswoman for the Boston mayor’s office.

The Massachusetts Department of Transportation spokeswoman Sara Lavoie said the agency’s snow removal efforts were aided by the fact that roads were largely clear of vehicles. Drivers, it seems, heeded Governor Deval Patrick’s pleas to avoid unnecessary travel on roads east of Marlborough.

“We will monitor post-storm conditions and watch closely to determine if we need to make another pass with salt and other de-icing materials,” Lavoie said. “We are always ready and we have sufficient amounts of salt on hand.”

MassDOT has long since spent more than the $43 million allocated in its budget for snow removal, and officials are still waiting for the state Legislature to grant supplemental funds that would increase the budget for this year’s winter cleanup to $93 million.

But at a MassDOT board of directors meeting last week, Highway Administrator Frank DePaola said he expected the costs to surpass even that amount. The jammed-up snow removal budget has rankled snow removal contractors, who typically receive their last payments for snow contract work in the summer.

“For a small business person doing what these folks have to do, it’s just not fair,” said MassDOT board of directors chairman John R. Jenkins at the meeting last week.

At Logan International Airport this afternoon, there were few stranded passengers in the check-in area. The airport had remained open throughout the storm, and extensive snow-plowing efforts early this morning kept the runways clear.

Still, 13 percent of flights were canceled Saturday night because of the storm, said Miraj A. Berry, a spokeswoman for the Massachusetts Port Authority.

“This meant that they did not begin with a full complement this morning, so they are working to accommodate those passengers,” Berry said.

Accommodating those passengers from canceled flights has been difficult, Berry said, because school vacation week means that most flights are already fully booked.

Mike Landsittel, of Weyland, said that Saturday’s storm has had no effect on his flight to Fort Meyers and the only hiccup has been leaving a piece of luggage at home.

“It’s been smooth,” Landsittel said. “Our flight’s on time, there wasn’t that much traffic, and it doesn’t look too crowded here.”