The Korean Centers for Disease Control and Prevention (KCDC) said that 86 people had tested positive so far in connection with the outbreak, but KCDC director Jeong Eun-kyeong said it was too early to call it a second wave.
The nightclubs in question do not advertise themselves as gay venues but have been described as such by local media.
Health authorities are continuing their efforts to curb coronavirus transmission by separating infected people from the public while juggling rights issues stemming from testing and contact-tracing methods.
The city has had to employ other measures to track them.
On Monday, an 84 year-old woman tested positive after she was in close contact with her a 29-year old grandson who was confirmed infected with the virus following clubbing in Itaewon.
The number of tests conducted on Monday nearly doubled to 6,544 from 3,500 a day earlier.
The cluster infection is believed to have started from a 29-year-old man who visited a total of five clubs and bars in Itaewon from the night of May 1 to the early hours of May 2. COVID-19 is the illness caused by the coronavirus. Those social distancing rules were lifted on May 6.
As part of a long-term fight against coronavirus, the KCDC will be given greater authority and renamed the Disease Control and Prevention Administration, Mr Moon said.
However, South Korean officials have not adopted large-scale restrictive measures, as the number of cases remained low.
Views on sexual minorities in South Korea have gradually improved in recent years, but anti-gay sentiments still run deep in the conservative country.
Health officials are struggling to trace "thousands of people" who went to clubs and bars in Itaewon, said Prime Minister Chung Sye-kyun. The other two were based in nearby Gyeonggi Province.
The group added that recent reporting would drive coronavirus cases underground, make "self-isolation hard and is a detriment to testing".
"Gays feel cornered. Either out yourself and potentially ruin your life, or stay in the closet but risk spreading corona", he said. The country tested 12,000 people Monday, up from a daily average of about 5,000, he said. The government reminded people to step up personal protection against the virus.
But South Korea is not the only apparent success story to report a regression recently. One case was in Daegu.
They reportedly say they did not visit the area near the nightclubs where the cluster infection was initially reported. "It's frustrating that the media is treating them all like sex clubs". "I would ask young people to refrain from reckless behavior that can lead our society to a crisis", he said in his speech on April 13.
Thus far South Korea has managed to handle the pandemic without a strict lockdown.
The new case cluster revived fear and frustration, suggesting a long, zigzagging path forward.
The clubber apparently did not wear a mask during his trips. A key component of that response was aggressive contract tracing, which has become hard with the Itaewon cluster because so numerous people exposed to the virus are afraid of being outed as gay. This meant that people continued to go out and even visit the party. "You need to otherwise you'll explode".
The authorities, with the backing of of 8,600 police officers, are also trying identify people from CCTV footage in the clubs.
"This is the time to muster courage", said Hong.
"We are living at the age of coronavirus and human beings (living in) coexistence".