The record of 13,012 comes just a day after the tally topped 8,000 for the first time amid a rapid spread of omicron despite the extension of tough social distancing rules.
Omicron is likely to account for more than 90% of new infections over the next few weeks, with the daily numbers surging to 20,000 to 30,000 or more, health officials said.
The variant became dominant in South Korea last week.
"Going forward, our top priority is to reduce critically ill patients and deaths," Prime Minister Kim Boo-kyum told an inter-ministry meeting on Wednesday.
A new testing policy has taken effect in four designated cities on a pilot basis, under which only priority groups can take a polymerase chain reaction (PCR) test while others should get a rapid antigen test first at a local clinic.
The programme will be expanded on Jan. 29 to include some 256 state-run testing stations nationwide, and then hundreds of local clinics starting Feb. 3, Kim said.
The government has also cut mandatory isolation for people who have been vaccinated but tested positive to seven days from 10, as part of efforts to free up resources for serious cases.
The ongoing surge has fuelled worries about chances of a new wave of infections ahead of the Lunar New Year holiday which begins on Saturday, when tens of millions travel nationwide to meet families.
South Korea, with a population of 52 million, has a tally of 762,983 infections and a death toll of 6,620, in what has largely been a COVID mitigation success story.
More than 95% of adults are fully vaccinated with some 58% having received a booster dose, according to the Korea Disease Control and Prevention Agency.