Eight boys and the coach remained inside the Tham Luang Nang Non cave as authorities paused the global effort to replenish air tanks along the treacherous exit route.
It comes a day after four of the boys were brought to the surface.
Thai Prime Minister Prayut Chan-o-cha paid a visit to all the rescue teams on Monday before leaving to see the evacuees at the hospital in the city centre.
"The divers that were assembled from many countries are proud to have conducted this operation until its success", Narongsak said.
A press briefing was to be held shortly.
At a news conference Monday, former Chiang Rai governor Narongsak Osotthanakorn said the second rescue involved numerous same divers who brought out the four boys on Sunday. "We will not push them too hard on their studies", he said, adding that extra tutoring will be given to allow the boys to catch up on their missed work.
The second phase of the rescue mission began around 11 a.m. local time (midnight ET) and took several hours.
CNN is reporting that eight of the twelve Thai soccer players have been rescued from the cave today.
Four of the 12 boys trapped undeground along with an adult were pulled out by divers yesterday.
"If Phra Pirun helps us, we might be able to do it very quickly", Narongsak said, again invoking the god of rain, who is widely revered in Thailand.
Update: 6:45 a.m. EDT- Rescue workers brought out a fifth boy at 5 p.m. local time (6 a.m. EDT) Monday from the cave in Thailand, the Guardian reported.
The boys became trapped in the cave on 23 June after heavy rains caused flooding, but were found alive last week by divers. Having completed this section, the boys are then handed over to separate, specialist rescue teams, who help assist them through the remainder of the cave, much of which they can wade through.
The divers also brought out touching notes for the families written by some of the trapped boys.
Thai officials have been tight-lipped about the rescue operation, and would not comment on how many people were removed Monday.
"You start to hallucinate, you see things, you hear things", said Dr. Robert Glatter, emergency physician at Lenox Hill Hospital.
But the death of a former Thai Navy SEAL diver who ran out of oxygen in the cave on Friday underscored the danger of the journey.
Four boys were brought out of the cave Monday and are "safe and conscious", Chiang Rai province acting Gov. Narongsak Osatanakorn said.
The extraction of the four on Monday followed a similar pattern to the previous day, with the youngsters emerging in quick succession just before nightfall after navigating a treacherous escape route of more than four kilometres.
Four more boys and their coach remain trapped inside the Tham Luang Nang Non cave complex.
Efforts to rescue the boys - aged between 11 and 16 - got going again on Monday after a break to replenish oxygen supplies and make other preparations deep inside the cave complex.
Change Mai governor Narongsak Osottanakorn said on Monday that the rescued boys' identities were being protected out of respect for the families of those still trapped inside the cave.
Hundreds of spectators and journalists have gathered in the area around the cave, hoping for news that the boys and their coach would emerge safely. They entered the complex with their 25-year-old coach following a practice - and before a sudden heavy rainstorm hit.