San Francisco Supervisor London Breed added to her narrow lead over former state Sen. Mark Leno in the tight race to become San Francisco’s next mayor, according to the latest batch of preliminary election results released Sunday afternoon.

Breed leads by 1,580 votes — 50.38 percent to Leno’s 49.62 percent — according to the latest update from the Department of Elections. She was up by fewer than 500 votes on Saturday, a lead of 50.13 percent to 49.87 percent. There are still 25,540 ballots left to count, the bulk of which should be processed by the department’s 4 p.m. update Monday.

“Obviously this is a nail-biter, but we’ve certainly been buoyed by the progress we’ve made over the last five days,” said Tara Moriarty, Breed’s campaign spokeswoman.

Breed overtook Leno in the preliminary ranked-choice election results for the first time Saturday, evoking elated gasps from a clutch of her supporters who gathered at City Hall to get a glimpse of the latest results in person.

“We remain cautiously optimistic,” Erin Mundy, Leno’s campaign manager, said Sunday.

The paper-thin lead Leno eked out after the first round of ranked-choice votes was tallied Wednesday morning has dwindled each day. Leno initially pulled ahead, thanks in large part to the staggering number of second-choice votes he picked up from Supervisor Jane Kim. Leno and Kim had urged their supporters to list them as each other’s No. 2 selection on the ranked-choice ballot.

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Kim all but conceded the race last week, telling her supporters on Facebook that she was “excited for a Mayor Mark Leno administration.”

The pace of ballot counting has been in keeping with the Department of Elections’ estimates before election day. The department got off to a slower start after election day, with updates on Wednesday and Thursday reflecting just over 12,700 processed ballots. But the tempo has quickened since then, with the department reporting results of around 20,000 ballots or more each day.

John Arntz, director of the department, has until July 5 to ensure that each valid ballot is counted, certify the election and officially declare a winner, but the outcome of the mayor’s race is expected to become clearer this week.

Tallying mail votes is a days-long process in San Francisco because each ballot is split into four separate cards, each of which has to be removed from a sealed envelope and hand-fed into vote-tallying machines.

“We’d be done counting most ballots if this election was a one-card ballot with only the mayor’s contest,” Arntz said.

Four-card ballots have become the norm for San Francisco elections, Arntz said. The city used a five-card ballot in the 2016 election.

“This is about as fast as we can work in relation to the various processes needed to verify and prepare ballots for counting, and with the equipment on hand to sort and then count the ballots,” Arntz said.

The city has four ballot-tabulating machines and a staff of 70 to 100 people opening envelopes and handling ballots each day.

A separate staff of around 40 people has been assigned to review the 14,000 provisional ballots turned in on election day. Provisional ballots are given to people who want to cast ballots but don’t appear on official voter registration lists, requiring an extra level of scrutiny when the time comes to process them.

They’re also given to people who vote by mail but didn’t receive a ballot, or don’t have their ballot with them and instead want to vote at a polling place, according to the California secretary of state.

Dominic Fracassa is a San Francisco Chronicle staff writer. Email: dfracassa@sfchronicle.com Twitter: @dominicfracassa