Chiang Rai province's acting governor, Narongsak Osatanakorn, who is in charge of the rescue, voiced confidence on Monday in the ongoing operation, provided the weather doesn't worsen.
Divers say the Thai dive and rescue is something that will go in the history books for divers around the world and rescuers will learn from for years to come.
The eight boys rescued after being trapped for more than two weeks were described as generally healthy.
The rescue team went into the cave to resume the operation at 11am (0400 GMT), Narongsak told a news conference earlier, adding he expected good news.
Hospital authorities said the first group of four boys rescued on Sunday were in good health and they were mulling letting their parents meet their wards from Tuesday but from a glass door separation. "When they're deprived of light and their comfort issues from home, there can be psychological sequelae that have to be addressed, and we've seen that before with people who have been held hostage or in kind of hard environmental situations", Ashton said, using a medical term for the after-effects of a traumatic incident.
The second evacuation attempt started at 11 a.m. local time (midnight ET) after rescue workers got some rest and refilled supplies.
The plight of the boys and their coach has drawn global attention since they became trapped when they went into the cave after soccer practice on June 23.
Four ambulances and a convoy of other vehicles arrived at the cave site Tuesday morning as rains hit the region.
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The names of the rescued boys have not been released out of "respect for those families whose sons are still trapped inside", the mission chief said.
The boys were found inside the cave by British rescue divers a week ago, about 4km (2.5 miles) from the cave mouth. According to media reports, rescue operations have been halted for the day. More worrying, however, oxygen levels in the chamber where the boys sought refuge were falling.
Four boys and the coach are still to be rescued. Scattered monsoon rains continued to risk percolating through the limestone cave walls to flood the tunnels with water.
Thai Navy Sgt. Saman Kunan, an ex-SEAL, died Friday from a lack of oxygen as he attempted to place air tanks inside the cave complex.
The eight boys who have left the cave are being treated in an isolation ward in a Chiang Rai hospital.
Rescuers move today to the entrance of the cave complex where five are still trapped in Mae Sai, Thailand.
A crack team of foreign divers and Thai Navy SEALS guided the boys during a nine-hour operation through almost 4 km (2.5 miles) of sometimes submerged, pitch-dark channels from where they have been trapped for more than a fortnight.
Reporters at the scene said three other boys had been extracted, but officials have yet to confirm that. All preparations, including replacing the compressed air cylinders positioned along the route out in the cave, take at least 20 hours, he said.
Narongsak said that experts told him new rain could shrink the unflooded space where the boys are sheltering to just 10 square meters (108 square feet).