In Vietnam, police said they had detained two people.
The discovery of the bodies in a container on an industrial estate has shone a spotlight on the illicit trade that sends the poor of Asia, Africa and the Middle East on perilous journeys to the West.
The alleged truck driver has already been charged over the deaths, and on Friday detectives said Eamon Harrison, 23, from Northern Ireland, was also accused of 39 counts of manslaughter as well as human trafficking and immigration offenses.
Harrison appeared at Dublin's High Court at the start of proceedings to extradite him from Ireland to Britain. He was remanded in custody until Nov. 11, a court spokesman said.
The bodies were found in the early hours of Oct. 23 after the container arrived in Britain from Zeebrugge in Belgium. The container was picked up at Purfleet dock in Essex, east of London, by a truck allegedly driven by Maurice Robinson, 25, from Northern Ireland.
The victims were found not long afterwards. Police have not confirmed the exact cause of their deaths.
They initially said the victims were thought to be Chinese, but on Friday evening they said they were now all believed to Vietnamese.
"We are in direct contact with a number of families in Vietnam and the UK, and we believe we have identified families for some of the victims," Essex police Assistant Chief Constable Tim Smith said on Twitter.