Moon said testing sites should operate longer hours to allow people working to get tested at their convenience and more drive-through testing facilities should be set up, Chung said.
The positive rate for the latest batch of tests was about 4.2%, compared to the year's average of 1.2%, according to the KDCA.
On Sunday, authorities said they will impose heightened social distancing rules for the capital Seoul and surrounding areas that will last until at least the end of the month.
South Korea avoided lockdowns but used an intensive system of tracing, testing and quarantining to tamp down two earlier waves of infection.
With this third wave, however, the government has faced increasing criticism as cases continue to rise despite measures such as mask mandates, curfews for restaurants and other businesses, and restricted public transportation.
Monday's total was down slightly from Sunday, when the agency reported 631 new cases, the largest daily tally since a peak in February and early March.
In total, South Korea has reported 38,161 cases, with 549 deaths.
Health authorities have said that if the current trend of cases continues, the hospital system could become overloaded.