Prime Minister Prayuth Chan-ocha visited the cave to inspect the rescue operation, with navy rescuers giving him a rousing cheer.
The flooded nature of the caves means that parts of the route of the dive are particularly unsafe, not only to the young boys, but also to the experienced divers, one of whom died last week while transporting oxygen tanks to the cave.
Last week, Musk tweeted that he was sending a team of SpaceX and Boring Company engineers to Chiang Rai in northern Thailand on Saturday to help rescue efforts, supplying technology such as Tesla Powerwall batteries and pumps for removing water.
FIFA president Gianni Infantino has invited the Thai boys' football team trapped in a cave to the World Cup final, as messages of support poured in from top players.
Police and military personnel use umbrellas to shield a rescued boy beng transferred on a stretcher from a helicopter to an ambulance after he emerged from the Tham Luang cave.
Two more boys are rescued from the cave.
"On behalf of the worldwide football community, I would like to join you in expressing my deepest sympathies and support to the families of the players and coach, as well as my solidarity with the people of Thailand at this time of great concern", Infantino wrote in a FIFA statement.
The remaining five members of the Thai soccer team will be rescued from the cave today, officials have confirmed.
But bringing out the remaining four boys and their coach could take more than one operation, Narongsak warned. But officials waited several hours before confirming their rescue. Tuesday's operation began just after 10am.
The boys were still being quarantined from their parents because of the risk of infection and would likely be kept in hospital for a week to undergo tests, officials said earlier on Tuesday.
The boys, ranging in age from 11 to 16, and their coach, 25, set out to explore the Tham Luang Nang Non cave system on June 23, but they became trapped after heavy rains quickly flooded passages.
Two of the boys had suspected lung infections but the four boys from the first group rescued were all walking around their beds.
Rescue workers in Thailand on Monday brought four boys safely out of a labyrinthine flooded cave complex where a 12-member soccer squad and their coach were trapped for more than two weeks, taking the total number rescued to eight.
He said a medic and three Seals in the cave, who have been looking after those trapped, will also come out.
The first boy rescued by Monday's team was freed around 4:30 p.m. local time and transferred to the same hospital as the first four boys who were rescued on Sunday.
Jedsada Chokdumrongsuk, permanent secretary at the Public Health Ministry, said the boys rescued Sunday were able to eat normal food by Tuesday, though they couldn't yet take the spicy dishes favoured by many Thais.
Rescue efforts bore fruit on Sunday as four of the children were pulled from the ordeal, while an additional four were added to the rescue tally on Monday.