Police said they believed the man found at Tilbury docks was Meet Singh Kapoor, from Afghanistan, who was 40.
Thirteen children aged as young as one were among the Afghan Sikhs rescued from the container on Saturday. Members of the 35-strong group found screaming and banging after arriving at the Essex port from Belgium have finished speaking to police and are now being cared for by Home Office officials, Essex police said. They are in the process of applying for asylum.
Inquiries will continue after a post-mortem examination carried out yesterday on Kapoor proved inconclusive.
The last four people remaining in Southend hospital, two adults and two children, left today.
Supt Trevor Roe said: "Tragically, one man died on the voyage and the investigation into his death is now being led by us. These people were found in an awful situation and our main priority is to look after them and ensure they are now safe following what would have been a horrendous ordeal."
Kamaljit Singh Mataharu, a translator used by police, told ITV News that the group believed they would have died if they remained trapped for another 30 minutes. He added that they told him: "It was pitch black, without any air. It soon became extremely uncomfortable."
Mataharu said there were 15 families, all from Kabul, packed into the container after they had travelled to Europe in a lorry. They told him they had been inside the container for 18 hours.
He said: "A little boy just said to me that he tried to wake his dad, but his dad didn't respond, and then they found out he was dead."
He had tears in his eyes while doing his translation work. "I normally never cry, but looking at these kids, how desperate they would have been to put their lives in such a state."
Essex police said the group included nine men and eight women aged between 18 and 72, as well as 13 children aged between one and 12.
Roe said: "The welfare and health of the people is our priority at this stage. We now understand that they are from Afghanistan and are of the Sikh faith. We have had a good deal of help from partners within the local Sikh community in the Tilbury area to ensure that these poor people, who would have been through a horrific ordeal, are supported in terms of their religious and clothing needs."
Belgian police earlier indicated that they believe the immigrants were probably already inside when the container was delivered to a European port before setting sail for Britain.
Bhai Amrik Singh, chairman of the Sikh Federation (UK), said: "It is a disgrace the persecution of the tiny minority of Sikhs from Afghanistan has largely been ignored and it takes an incident like this to remind us all that they are also being exploited by human traffickers."
Nick Alston, police and crime commissioner for Essex, said: "The death of the man in a container found at Tilbury on Saturday was a shocking incident. One person died and many others might very easily have done so."
He said the developments should "highlight for us the evil of the abuse and exploitation of the vulnerable," adding: "The police and partner agencies are continuing to investigate the circumstances of those found in that container. But we can be sure that almost all cases of human trafficking involve exploitation by callous organised criminal gangs."
Footage obtained by ITV News apparently shows members of the group after they were released from the container. The clip shows some of the migrants sitting on the floor while the sound of wailing and shouting can be heard.
A Home Office spokesman said all the people involved were now in the process of claiming asylum in Britain. He said: "We are continuing to provide expert support, along with the National Crime Agency, to the Essex police investigation into the discovery of 35 people in a container at the port of Tilbury on Saturday.
"This tragic incident is a reminder of the devastating human consequences of illegal migration and we will do all we can to help bring those responsible to justice. Following the conclusion of police interviews, the individuals involved have now been passed into our care. All 34 are now in the process of claiming asylum in the UK and we are providing accommodation and support to those who require it while their cases are considered.
"The UK takes its international obligations extremely seriously and has a proud history of offering protection to those who need it."
Die Menschen,  die am 16. August im Hafen von Tilbury in einem Schiffscontainer entdeckt wurden, kommen tatsächlich, wie von ihnen selbst angegeben, aus Afghanistan. Es handelt sich nach Angaben der Polizei um 15 Familien aus Kabul, die der Sikh-Religion angehören. Auch der Tote, der gefunden wurde, konnte identifiziert werden; seine Todesursache ist aber noch immer Gegenstand einer Untersuchung.

Da es in Afghanistan nur noch eine sehr kleine Gruppe von Sikhs gibt, hatte es zunächst Zweifel an der Behauptung gegeben, es handele sich um Afghanen, und man hatte sehr viel mehr eine Herkunft aus dem indischen Bundeststaat Punjab vermutet. Die afghanischen Sikhs sind Diskriminierungen im Alltag ausgesetzt, was ihre Zahl ständig sinken lässt und sie in die Hände von Menschenschmugglern treibt. Dies ist aber ein weitgehend unbekannter Umstand, wie Bhai Amrik Singh, der Vorsitzende der Sikh Federation der Interessenvertretung der Sikhs in Großbritannien, sagte. In Indien, das zunächst als Ursprungsland der Flüchtlinge vermutet wurde, sind die Sikhs im Gegensatz zu Afghanistan vollständig in die Gesllschaft integriert. Die in Tilbury entdeckten Menschen werden nun von den örtlichen Behörden als Flüchtlinge versorgt, und ein Asylverfahren wurde eröffnet.