The global effort to save the group has paired divers in "buddy teams" with the remaining eight boys and their coach.
Rescue mission chief Narongsak Osottanakorn said it had resumed, with divers re-entering the cave at 11am local time (4pm NZT).
"We've been working continuously overnight", a Chiang Rai government source told AFP on Monday morning, requesting anonymity, and confirming that there had only been a pause of the actual extraction operations. It was unclear who was inside the ambulance or the helicopter.
Interior Minister Anupong Paojinda had said early Monday that the same group of expert divers who took part in Sunday's rescue would return to extricate the others because they know the cave conditions and what to do.
Four boys were successfully rescued from the Tham Laung cave area on Sunday in a delicate and lengthy operation. Its eighth floor has been reserved for the soccer team, their coach and their families - who have maintained a vigil by the cave's entrance while the boys have been underground.
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. Eight people remain trapped in the flooded cave system. With the assistance of divers, they swam out with oxygen tanks, along a 2-mile long rope guiding them out.
The 12 boys - ages 11 to 16 - entered the massive cave complex on the frontier of Thailand and Myanmar with their 25-year-old coach on June 23.
After his visit, he went to Chiangrai Prachanukroh Hospital to visit the rescued boys.
Relatives said the boys had been inside the labyrinthine complex during the dry season.
"It's very likely that while the boys were in the cave but not yet discovered by rescuers that they experienced various degrees of anxiety, fear, confusion, vulnerability and dependency, and perhaps hopelessness", said Paul Auerbach, of the Department of Emergency Medicine at Stanford University's medical school.
"The next step is to make sure those kids and their families are safe because living in a cave has a different environment, which might contain animals that could transmit any disease", a hospital statement said.
"I'm hoping for good news today", he said. Visitors would need to wear sanitized clothes and stay two meters away from the children. He said that process can take several hours.
Four boys and their soccer coach remain in the cave. Two other boys came out at 7:40 p.m. and 7:50 p.m.
"Adolescents are especially social creatures, and having friends with them as well as their coach would be a tremendous help", said David Spiegel, a professor of psychiatry and behavioral sciences at Stanford University's medical school.
The remaining nine boys are still at the location where divers had found them about 500 metres from the so-called Pattaya Beach.
The rescue mission has been a huge operation, led by the Royal Thai Navy's SEAL unit, and supported by a cast of hundreds.