Rescue operation commander Narongsak Osottanakorn declined to say how many people would be taken out on Monday, but a Thai navy source said it was likely to be four or five.
Thai rescue team members walk inside a cave where 12 boys and their soccer coach have been trapped since June 23.
The four boys guided from the cave Sunday in an urgent and risky operation that involved them diving through the cave's dark, tight and twisting passages were happy and in good health, authorities said.
The rescued children were taken to the hospital, and officials said early Monday that they are strong and safe but need to undergo detailed medical checks. Thai Prime Minister Prayut Chan-ocha was scheduled to make a second visit to the scene late on Monday and the president of soccer's governing body, FIFA, Gianni Infantino, today invited the boys to the World Cup final in Moscow on Sunday if they make it out of the cave in time. The second operation started at 11 a.m. local time Monday (midnight ET). Monsoon flooding cut off their escape and prevented rescuers from finding them for nearly 10 days.
Families have not been told which boys are free, so all parents have agreed to remain at the cave until the entire team is rescued.
Their escape led to an explosion of jubilation on social media in Thailand as the rescued boys were rushed to hospital.
It was not immediately clear Monday how the overnight rains had impacted water levels inside the flooded cave.More news: Oil Prices Drop On US-China Trade War, Saudi Production Boost
The death Friday of a former Thai navy SEAL, Saman Gunan, underscored the risks. Their families were being kept at a distance because of fears of infection and the emaciated-looking boys were eating a rice-based porridge because they were still too weak to take regular food, authorities said.
"They are all of our students, all of our friends, all of our children", she said.
Thirteen foreign and five Thai divers are taking part in the rescue operation.
The operation began at 11 am (local time) and took nine hours, two fewer than Sunday, with rescuers helped by the experience they've accumulated, Narongsak told a news conference punctuated with rounds of applause.
The risky bid to rescue the boys - aged between 11 and 16 - got going again hours earlier on Monday after a break to replenish oxygen supplies and make other preparations deep inside the cave complex. Heavy rain has struck the region intermittently over the last three days and further downpours could set back draining efforts at the cave.
Phuwadech said he remains anxious about Tun and the others, particularly after seeing them in videos Thai Navy SEAL divers have taken of the boys inside the cave and posted on Facebook. "And we have more expertise than yesterday", he said. I've told Yod to get ready to take me out for fried chicken.
Thirteen medical teams were stationed outside the cave - each with its own helicopter and ambulance - one for each of 12 boys and their coach. They were quickly transported to a hospital in the town of Chiang Rai, the provincial capital.