Nargonsak said this phase may take longer than the previous two rescue missions, the longer, first missing taking 11 hours.
Parents of the four boys rescued on Sunday were allowed to see them through a glass window at the hospital, public health officials said on Tuesday, but they will be quarantined for the time being.
A doctor and three Thai Navy SEALs, who have stayed with the group since shortly after they were discovered huddled on a muddy ledge deep inside the Tham Luang cave complex, will also come out, Narongsak added.
19 divers are said to be involved in the operation, which began at just after 10:00am local time.
Thai Navy SEALS divers have already rescued eight of the boys.
They were trapped in the Tham Luan Nang Non cave that became flooded by monsoon rains while they were exploring it after a soccer practice on June 23.
They were first found by British divers more than a week ago on Monday after spending 10 days with limited food or water.
The eight boys undergoing treatment at Chiang Rai Prachanukroh Hospital were in general good health, both physically and mentally, he said.
It is believed that the second lot of boys are in better health than the first four who were brought out first because they were deemed the "weakest".
Jesada said they were uncertain what type of infections the boys could face "because we have never experienced this kind of issue from a deep cave".
"They wrote: "[Manchester United] is relieved to learn that the 12 footballers and their coach trapped in a cave in Thailand are now safe. Their body temperatures were very low when they emerged, and two are suspected of having lung inflammation.
The four boys who were recovered on Sunday are now eating normal food, Jedsada said.
Sadly, the boys and their coach had to decline under orders from their doctors to remain in the hospital for seven days for monitoring.
Heavy rains lashed the northern Thai region late Monday and a steady downpour has continued Tuesday.
Many of the Wild Boars players came from impoverished families, and Chanthawong reportedly helped put in place a system whereby kids would get soccer gear from sponsors in return for good grades.
The third and hopefully final rescue mission is now underway.
Elon Musk visited the flooded Thailand cave to offer a "kid-sized" submarine to help with the rescue - only to be told his technology wasn't "practical" to the mission.
But he added rescuers may need more than 20 hours, to replan and replenish oxygen supplies, to be ready to mount the next mission.
Celebrations spread across the world as the last of the Wild Boar football team were brought out of the caves following a three-part rescue mission that spread over several days. "Leaving here in case it may be useful in the future", he tweeted Monday.