The operation on Tuesday moved quickly, raising hopes that all 12 boys and their adult coach from the Wild Boars soccer team would be at the surface by the end of the day.
All eight boys rescued on the first two days are being treated in an isolation ward in a Chiang Rai hospital.
The rescued boys are still being held in quarantine while lab tests are being done and have not been able to have direct contact with their parents yet.
Eight boys had been taken out in the past two days - four on Sunday and another four on Monday.
On Sunday, officials said it could "take days" to rescue the remaining boys and their coach.
It's been more than two weeks since the 12 boys and their soccer coach became trapped in the flooded Tham Luang cave at the Khun Nam Nang Non Forest Park in Chiang Rai.
Musk had offered mini submarine, named Wild Boar after the kids' football team, to help in the rescue operation.
"Today we might have to wait longer, but it will be worth the wait", he added.
The rest four boys and their coach that are still trapped in the cave will be pulled out in a final mission that has resumed on Tuesday.
Thirteen foreign divers and five members of Thailand's elite navy SEAL unit make up the main team guiding the boys to safety.
When asked about his first reaction as he saw the first set of boys who were rescued he admitted that he was "very scared".
Chanthawong has reportedly taught meditation to the boys to keep them calm in the cave, and is suffering from malnutrition because he gave his food rations to the young players.
The eight boys brought out on Sunday and Monday were in good health overall, although two had suspected lung infections.
Narongsak Osottanakorn told a news conference the latest rescue operation began at about 10:08 am local time.
"Everyone is in a good mental state", he said but added psychologists will work with them.
Additionally, there was no heavy rain near the caves Monday morning, according to Jonathan Erdman, weather.com meteorologist, creating a break for rescuers who once feared the worst from monsoon rains.
People across Thailand cheered the rescue operation, including at the Mae Sai Prasitsart school where six of the trapped boys are students. "We can see that everything is okay as they're eating well", said Thongchai Lertwilairattanapong, Inspector General of the Public Health Ministry.
Some family members have been allowed to visit the boys in hospital, but are only allowed to observe them from behind a glass window. He said he left the equipment there in case rescuers could use it in the future.