The Liverpool bomber attempted to launch a fresh legal appeal to stay in the UK in January and then began making purchases for his attack in April, it has emerged.
Counter-terror police said Iraq-born Emad Al Swealmeen rented a property in the city seven months ago.
He had suffered from periods of mental illness which will “form part of the investigation”, they added.
Earlier police extended a cordon at an address linked to the asylum seeker.
Christian convert Al Swealmeen, 32, was a passenger in a taxi when his homemade device exploded outside Liverpool Women’s Hospital shortly before 11: 00 GMT on Remembrance Sunday.
A post-mortem found he died from injuries caused by the explosion and fire.
The driver David Perry escaped seconds before his car was engulfed in flames and has since been discharged from hospital.
Details of Al-Swealmeen’s immigration legal history provided to the BBC by court officials show he had a long and complicated series of applications and appeals to remain in the country.
It remains unclear whether the Home Office tried to remove him after his first failed application to stay more than six years ago and why his final appeal was unresolved.
According to court records provided by officials, Al Swealmeen was first refused asylum in 2014 and also lost further appeals in 2015.
The records show in August 2015 he began seeking to convert to Christianity and adopted a new name, Enzo Almeni, as part of the conversion.
They also document that he renewed his immigration appeal under this alternative name in January this year.
His submissions were under review at the time of his death and the Home Office has refused to discuss the case with BBC News.
Police said Al Swealmeen had lived at a property in Sutcliffe Street in the Kensington area of Liverpool for some time and began renting a property in Rutland Avenue, near Sefton Park, in April.
Assistant Chief Constable Russ Jackson, head of Counter Terrorism Policing North West, said “significant items” had been recovered from the Rutland Avenue address, where searches have been taking place.
The cordon was extended around Sutcliffe Street on Wednesday afternoon after several suspicious packages were found, Counter Terrorism Policing North West said.
But the cordon was retracted hours later to only cover Sutcliffe Street.
The public was warned bomb disposal teams may be present in the area on Thursday, but this will “be solely out of caution in case further items are found,” a spokesman added.
Nearby homes were not evacuated but residents living within the cordon were told they could not return home.
A bomb disposal vehicle attended the scene, along with a fire engine and four police vans.
Home Secretary Priti Patel said the UK terror threat level had been raised from “substantial” to “severe”, meaning an attack is “highly likely”, because the explosion in Liverpool was the second incident in a month.
Assistant Commissioner Matt Jukes, the most senior counter-terrorism policing officer in the UK, has urged the public to be vigilant in the run up to Christmas.
He told the BBC that in the past terrorist attacks have led to other attacks and that is why the threat level has been raised.
Sheikh Salem Al-Suwailmieen, one of Al Swealmeen’s extended family, said in a statement the suspected bomber’s mother was from Iraq and confirmed he was born there.
He said he lived in the United Arab Emirates (UAE), before going to Syria then Turkey and eventually ending up in the UK where he applied for asylum on the grounds that he was Syrian.
Al Swealmeen had suffered from mental illnesses, he added.
Mersey Care NHS Foundation Trust, which provides mental health services, said Al Swealmeen had received treatment in the past but was not under their care at the time of the explosion.
Mr Jackson, head of Counter Terrorism Policing North West, said: “A complex picture is emerging over the purchases of the component parts of the device, we know that Al Swealmeen rented the property from April this year and we believe relevant purchases have been made at least since that time.”
He said police were “not finding any link to others in the Merseyside area of concern but this remains a fast moving investigation and as more becomes known we cannot rule out action against others”.
On Monday evening, police released without charge four men who had been arrested under terrorism laws in Liverpool in connection with the attack.
Malcolm Hitchcott, a Christian activist in Liverpool, said he and his wife offered Al Swealmeen a place to stay for eight months in 2017 after he told them he had lost his case and was destitute.
“He was absolutely genuine, as far as I could tell. I was in no doubt by the time that he left us at the end of that eight months that he was a Christian,” Mr Hitchcott told BBC Radio Merseyside.
Al Swealmeen was baptised at Liverpool Cathedral in 2015 and confirmed in 2017 but the Diocese of Liverpool’s communications director, Stuart Haynes, said they lost contact with him in 2018.
The Church of England has said it is not aware of any link between conversion and asylum system abuse after newspaper reports suggested changing religion could be a way to “game the asylum system”.
Liverpool Cathedral said it had “developed robust processes for discerning whether someone might be expressing a genuine commitment to faith”.
The Reverend Mike Hindley said Al Swealmeen was also involved in the Emmanuel Church in the Fazakerley area of Liverpool from 2017 to 2019, when they lost touch with him.
He said Al Swealmeen, who parishioners knew as Enzo, “never made a secret” of the fact he had mental health problems but did not go into great detail about them with him.
A spokesman for the City of Liverpool College said Al Swealmeen was an adult cake decorating student at the college during the 2018-2019 academic year.
Joy Gambardella, a lay reader at Emmanuel Church, said he “used to make cakes for the church and sell them”.
Praising her “amazing” and “resilient staff” for their response to the “traumatic” incident, Liverpool Women’s Hospital chief executive Kathryn Thomson said she had “no knowledge” of whether Al Swealmeen had any previous connection to the institution.
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