Thongchai said one member of the final group of four boys and the coach who arrived at the hospital on Tuesday evening had a slight lung infection.
The youngest, 11, appeared asleep under a crisp white sheet while others, including their 25-year-old soccer coach, sat in bed, their faces obscured by green surgical masks.
Narongsak said worldwide rescuers led the final phase of the unsafe mission.
The hashtag #Hooyah was hugely popular on social media with people showing their support for the hundreds of rescuers, including divers from around the world, who helped to get the boys out. They were found by a pair of British divers 10 days later, huddled on a small, dry shelf just above the water, smiling with relief but visibly skinny.
The first four boys are healthy and can eat regular food, Thongchai said at Monday's press briefing, adding that those in the second group suffering from lung inflammation were getting better.
"From our assessment, they are in good condition and not stressed", Thongchai said.
Thailand's junta chief told reporters on Tuesday that the group had been given a "minor tranquiliser" to help calm their nerves.
He examined the boys in the cave and decided when they were well enough to be rescued.
The rescue efforts cost the life of one rescue worker - a former Thai Navy Seal ran out of air inside the cave system. Before their discovery, they survived by drinking water dripping into their cramped refuge.
The search started immediately but rising waters made the effort hard.
Thailand spent yesterday celebrating the successful mission.
Health officials said the boys will have a mental health evaluation to address any problems caused by their ordeal.
Elsewhere, Duangduen Sittiwongsa, a classmate of 16-year-old Pheeraphat "Night" Sompiengjai, whose birthday fell on the same day the team entered the cave, said they would give him cake and gifts when he returned to school.
Seeing his son through the glass, he said he only wanted to hug him.
Rescuers had weighed up several options to save the boys, including keeping them in the cave through the months-long monsoon season.
But they were prodded into the risky task of "diving out" the team through submerged chambers and claustrophobic passages as oxygen levels in the cave plummeted and rain menaced.
Cheers erupted at a local government office where dozens of volunteers and journalists were awaiting news of whether the intricate and high-risk rescue mission for the final boys stranded in a Thai cave network had succeeded.