Ivan Karadzic, a Danish dive instructor who lives in Thailand and who was involved in Sunday's rescue effort as a support diver, said that the operation "went surprisingly well, we were expecting bad things to happen, and they didn't".
Officials last week said they would bring the fittest people in the group out first, but Narongsak later said that whoever was ready first would be escorted out.
A fifth boy was rescued on Monday afternoon, a Royal Thai Navy official said.
He said the plan is to launch the third rescue mission within 20 hours, and he is hoping for "100 per cent success". Last Monday they were discovered huddled on a narrow rock shelf deep within the flooded cave system.
All four boys brought out of the cave on Monday were taken by ambulance to a waiting helicopter which ferried them to hospital.
Rescuers have been navigating a unsafe and complicated plan to get the children out under the threat of heavy rain and rising water underground.
Authorities said the next phase could begin any time within a 10-hour window that began about 0000 GMT (7am local time) on Monday. CNN's Jonathan Miller reported that the eight boys and coach who remain stranded in the cave would likely have to stay put for a couple of days due to issues restocking oxygen supplies.
The anxious families say they have forgiven 25-year-old soccer coach Ekapol Chanthawong who led the boys on what was supposed to be an innocent adventure in cave exploring more than two weeks ago.
According to the Associated Press, a Thai official revealed that the healthiest boys were taken out of the cave first.
There was no word on the condition of any of the people brought out on Monday.
Kanet Pongsuwan, the principal of the nearby Mae Sai Prasitsart School - which six of the boys attend - said that the rescued students will be eased back into the classroom.
The rescued boys are being kept in sterilized isolation units at a nearby hospital, which prohibits physical contact, and their parents have not been told they're safe.
"The four children are well at Chiang Rai Prachanukroh hospital".
"The rescue has been vast", said volunteer helper Somjit Saenset, 56. Four boys and their football coach are still trapped inside the cave.
"We have not been told which child has been brought out ..."
Paramedics with a stretcher believed to be carrying one of the boys from the soccer team in Chiang Rai on Sunday.
"I'm hoping for good news".
Soldiers arrive outside the Tham Luang cave complex in the northern province of Chiang Rai, Thailand, July 8, 2018. Soldiers, medics, engineers and volunteers in yellow shirts milled around but the mood was relaxed.
After reaching the area where the boys were trapped, the divers used a rope, which was connected to the outside of the cave, to guide them through the exit way.
'Doctors have told us we must be careful about the food that is given to them because they are fragile after starving for many days.
The cave complex is off-limits during the rainy season, which usually runs from May to October, when downpours can quickly flood it.