The "Wild Boars" football team and their coach got trapped on June 23 when they set out to explore the vast cave complex after football practice and the tunnels became flooded.
Eight boys are still inside the cave and along with the team coach.
It took nine hours Sunday to rescue the first two boys, who were described as the weakest of the group.
Four of the 12 boys were rescued in the first phase of the operation.
Interior Minister Anupong Paochinda told reporters the four rescued boys rescued on Sunday were in good health in hospital but did not give details.
That makes for eight of the 13 people who have been trapped under increasingly unsafe conditions since late last month. Crews should replenish air tanks along the route before rescuing the others.
Finally, the 13 were discovered deep inside the mountain, fleeing rising waters and depleting oxygen levels.
Authorities have said the mission could take up to four days to complete and heavy rain is forecast in the coming days.More news: What they're saying about Cousins and Warriors one-upping LeBron and Lakers
Thai premier Prayut Chan-O-Cha was due to visit the rescue operation on Monday. "I've been involved in cave rescue for 30 years and I can not even think of one that was more complicated".
Workers have been laboring around the clock to pump water out of the cave, and authorities said Monday that heavy downpours overnight did not raise water levels inside.
The rescued boys were being treated by medics at the field hospital, and were later airlifted to a larger nearby hospital. Officials said Monday that despite rain Monday morning, the weather was holding out so far. If the tests were successful, the sub was to be placed on a 17-hour flight to Thailand.
The most unsafe part of the journey out of the labyrinth cave system is the first kilometer, during which they are required to squeeze through a narrow flooded channel.
A spokesman for Musk's Boring Co. said Sunday that the company has four engineers who are "offering support in any way the [Thailand] government deems useful". With the assistance of divers, they swam out with oxygen tanks, along a 2-mile long rope guiding them out.
Authorities declined to say how many boys would be extracted on Monday.
During the press conference, Thai authorities said they were hoping it wouldn't rain for the rest of the rescue operation so that the process could be faster.
Handwritten notes by the children trapped in a cave read, from the top: 'Please don't worry Dad and Mom, Biw has been gone for two weeks, I will help mom sell her goods every free day, I'll quickly make my way.' The message below reads: 'Please don't worry, be happy.
But with oxygen levels inside dropping to unsafe lows and the prospect of heavy rains flooding the area completely, authorities decided they had to move quickly and take the group out through the water-filled tunnels.