The burden of bringing out five people - having extracted four on Sunday and four more on Monday - will stretch the capacities of the rescue team.
Family members have seen at least some of the boys from behind a glass isolation barrier, and Mr Jesada said doctors may let the boys walk around their beds on Tuesday.
Thailand has been riveted by the dramatic rescue mission to save the "Wild Boars" team after they first got trapped in Tham Luang more than two weeks ago by rising floodwaters.
The plight of the boys and their coach has drawn worldwide attention since they became trapped when they went into the cave after soccer practice on June 23.
Divers have rescued eight of the boys. The hands form a circle, symbolizing the bond and the joint effort by several worldwide teams to rescue the boys.
"If nothing goes wrong the four boys and the coach, along with the doctor and three Seals with them, will leave the cave together today".
Narongsak said he was grateful for Musk's support but the equipment was impractical for the rescue mission.
"They (all the boys) will have to stay in the hospital for one week to wait for their results and to see if anything changes", he said.
The chief the rescue mission said they are being kept apart from relatives because of infection concerns, Reuters reports.
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Musk also said that a second sub, which is apparently 30cm shorter than the first one designed, was nearly complete, meaning that more than one device would be available for the rescues.
Four more of the boys were carried on stretchers out of the labyrinthine Tham Luang cave on the Myanmar border at dusk on Monday, bringing to eight the number brought out after two rescue pushes on successive days.
But two of the boys freed from the partially flooded Thai cave have suspected lung infections, following their 16 day ordeal underground.
Nineteen divers are involved in the rescue attempt and officials said water levels remain stable, unaffected by the rain.
Mr Musk and his team have been in touch with officials in Thailand, a SpaceX spokesperson confirmed, and Musk tweeted Sunday that the sub's design is based on feedback from dive teams assisting with the rescue operation. After they were found on July 2, officials cautioned it could take some time to get them out, but with heavy rain forecast to hit the region authorities made a decision to act.
Chiang Rai acting Gov. Narongsak Osattanakorn said the second phase began at 11 a.m. and authorities "hope to hear good news in the next few hours". "And we will do it faster because we are afraid of the rain".
The four boys pulled from the cave Sunday in an urgent and unsafe operation that involved them diving through the cave's dark, tight and twisting passages were happy and in good health, authorities said.
The boys and the coach were found more than a week after they became stuck on a small, muddy patch deep in the cave's network of chambers.