Of the dead, 37 were women and 17 were men.
Police say the driver failed to turn on the air-conditioning in the back of the truck.
He stopped driving when he heard people banging on the walls and fled after opening the door to the container and seeing the bodies.
Kraithong told the AFP news agency that the migrants "tried to bang on the walls of the container to tell the driver they were dying, but he told them to shut up as police would hear them when they crossed through checkpoints inside Thailand".
Police said on Thursday that more than 20 of the 67 survivors had been hospitalised, with the rest detained for questioning.
The migrants reportedly each paid a smuggling ring $157 to be taken to Phuket to find jobs.
According to Thailand's labour ministry, about 540,000 migrant workers mostly from Myanmar are registered to work in the country.
But rights groups claim that as many as a million undocumented workers could be working in Thailand and are often subjected to exploitation by employers.
Tens of thousands of people also flee persecution and poverty in Myanmar, with many ending up in Malaysia where they are not given refugee status and are classified as illegals.