The BBC has spoken to the families of three Vietnamese people who are worried their relatives may have been in the trailer.The family of one woman say she sent a text saying she was suffocating.
They say Pham Tra My, 26, sent the message on Tuesday night and they have not been able to contact her since. They said they had paid £30,000 for her to be smuggled to Britain.
Two other families have also been in touch with the BBC. They are relatives of a 26-year-old man and a 19-year-old woman.
The 19-year-old’s brother said she called him early on Tuesday to say she was getting in to a container and was turning off her phone to avoid detection.
There has been no word from her since, he said, but a people smuggler had returned money to the family.
Relatives of the 26-year-old – with whom she was said to be travelling – also received money back, according to the younger woman’s brother.
Essex Police initially said the victims – 31 men and eight women – were believed to be Chinese.
They were found at an industrial estate in Grays at 01:40 BST on Wednesday.
Detectives are still questioning the lorry driver on suspicion of murder.
Police have been given extra time to question driver Mo Robinson, of County Armagh in Northern Ireland, who was arrested on Wednesday.
Post-mortem examinations are due to start later after the first 11 bodies were moved from Tilbury Port to Broomfield Hospital in Chelmsford.
Private ambulances continued transporting more of the 39 bodies from the refrigerated lorry trailer to the mortuary on Friday.
The trailer arrived in Purfleet on the River Thames from Zeebrugge in Belgium at 00:30 on Wednesday.
It left the port shortly after 01:05 the same day and the bodies were found in the trailer at Waterglade Industrial Park about 30 minutes later.
Police said recovering the bodies would take time and the dignity of the victims was its primary concern.
The Chinese Ambassador to the UK, Liu Xiaoming, said he had sent a team to Essex to help verify the identity of the victims. He added that their nationality was yet to be confirmed.
Essex Police believes the lorry arrived in Holyhead in north Wales on Sunday, having travelled from Dublin.
Global Trailer Rentals Ltd told RTE News it owned the trailer and said it had been hired on 15 October.
Tracking data from the trailer shows it had travelled between cities in Belgium and France, including Dunkirk, Bruges, and Lille, in the days before the discovery, sources said.