The aide, Sitthichai Klangpattana, didn't comment on the boys' health or say how well the operation has gone.
Sunday's rescue mission was described as a success by Thai officials.
The ambulance drove toward a helipad, where a helicopter was seen taking off shortly after to the cheers of the crowd below.
Authorities have been tightlipped about the progress of Monday's operation.
The boys set out to explore the vast Tham Luang cave, which is known in the area as off-limits, after soccer practice on June 23.
The fifth boy was taken by helicopter to Chiangrai Prachanukroh Hospital in the provincial capital Chiang Rai, while the sixth remained inside a medical tent before being moved. Medics appeared to remove one person on a stretcher but hid the person's identity behind multiple white umbrellas. However, the rescue effort resumed a few hours earlier than planned on Monday.
Four more young footballers and their coach at the Mu Pa (WIld Boar) Academy Mae Sai, Ekkapol Chantawong, are still on the ledge.
It is understood the same divers who rescued the first four boys have been given the job of getting the remaining team members out.
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"Everything is all good - weather, water and the boys' conditions", he said, adding, "We will speed up [the operation]". It takes several hours. Still, they were undergoing medical checks in a hospital and were not yet allowed close contact with relatives due to fear of infections, who were able to see them through a glass partition.
In a letter of his own, the coach, Ekapol Chanthawong, apologized to the boys' parents for the ordeal.
"The water level is low".
Thailand officials have not released the names of the boys rescued, so as not to upset the parents of any child still trapped.
The mission was paused overnight on Sunday for air tanks to be replaced.
There was no word on the condition of any of the people brought out on Monday. The outlet reported that eight boys remain inside the cave.
At the hospital in Chiang Rai, green canvas sheets had been put up to block the entrance from view.
Thailand's Meteorological Department said there was a 60 percent chance of rain Monday with thunderstorms forecast throughout the week. If successful, the sub would be placed on a 17-hour flight to Thailand. Four came out Monday, and it is possible Tuesday's plan was to bring out four more.
The submarine is small and light enough to be carried by two divers, and can navigate through the narrow submerged areas in the cave, which is said to be the biggest challenge in rescuing the children trapped in the cave. Divers removed another boy about an hour later. They were quickly transported to a hospital in the town of Chiang Rai, the provincial capital.
The boys wore full diving face masks as a team of diving experts guided them along a 4.7-kilometer (2.9-mile) route out of the cave complex. It is a race against the clock with heavy rain expected in coming days, which would again dangerously flood the tunnels.