According to our correspondent, Thai officials have warned the rescue could take three or four days in the worst case scenario.
Narongsak said that there was no time limit on the rescue and that its progress would depend on weather and conditions inside the cave, but expected it to last through Monday.
The rescue mission began after rain showers soaked the Tham Luang Cave area in northern Chiang Rai province for the past 24 hours, heightening the risks in what the governor has called a "war with water and time" to save the team.
Two other boys emerged from the cave at 7:40pm and 7:50pm local time.
A former Navy SEAL and combat swimmer said that the risky and harrowing rescue of a soccer team trapped in a flooded Thailand cave was comparable to climbing the tallest mountain in the world. Thirty-five emergency doctors were on standby as the "exhausted" boys were flown by military helicopter or driven in ambulances to Chiang Rai Prachanukroh hospital.
Two divers will accompany one child as they stick to a cave line that stretches the entire mile and a half distance.
The boys, ages 11-16, and their 25-year-old coach have been trapped in the complex cave system since June 23 by flooded passageways.
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Mr Ekapol Chanthawong, his assistant, was to take the younger boys to a soccer field nestled by the Doi Nang Non mountain range, a formation with numerous waterfalls and caves that straddles the Thai-Myanmar border. The Thai official in Sunday night's news conference said rescue workers need "about 10 hours" to prepare for next operation.
U.S. President Donald Trump tweeted Sunday morning, "The U.S.is working very closely with the government of Thailand to help get all of the children out of the cave and to safety".
Earlier on Sunday all media and non-rescue personnel were cleared from the cave site area to facilitate space for rescue personnel and medical team. The first four rescued youths surfaced around 12pm GMT (7am ET) and were immediately taken to hospital to receive treatment.
After pumping out water for days, experts say the window to rescue the boys is closing and it is now or never.
"They insist they are ready to come out", Narongsak said Sunday of the rescue team.
The next phase of the operation to rescue the remaining eight players and their coach will begin in 10 to 20 hours, commanders said, giving operators time to replenish supplies and evaluate the rescue plan and weather.