Richard Walsh, 43, was awaiting trial accused of attempting to murder two schoolboys in Hampshire when he was found hanged in his cell.
An inquest jury found he should have been detained in hospital rather than held in Belmarsh Prison.
Southern Health NHS Trust said it would ensure better "sharing of information".
On 27 September a verdict of suicide, contributed to by neglect, was recorded at the inquest into Mr Walsh's death.
Mr Walsh was held at Waterlooville police station following the stabbing of two boys, aged 12 and 13, in Southleigh Road in Havant on 26 June last year.
Southwark Coroner's Court heard he was "saying strange things", became "delusional", was "obsessed with a woman police officer", and hid under a blanket when questioned.
He was assessed by two independent psychiatrists and a mental health professional approved by Hampshire County Council, but was not sectioned.
The inquest jury said there had been "gross failures of medical care" as they failed to interrogate the custody record and the custody officer and did not speak to Mr Walsh's family or GP.
"Had they done, then Mr Walsh would have been detained in a hospital.
"It is more likely than not... that if Mr Walsh had been detained in a hospital... he would not have killed himself," the jury found.
Mr Walsh was found hanged on 19 July 2015, the day before he was due to appear at Portsmouth Crown Court.
Following the ruling, Southern Health's medical director Dr Lesley Stevens said: "We fully accept the jury's findings and are acting quickly to improve assessments and communication between health organisations and the criminal justice system."
The trust said it would delay assessments of detainees if custody logs were not available and would implement "clear and robust processes" for sharing information between clinical teams in the custody and prison services.
A Prison Service spokesman said it had already taken action following a Prisons and Probation Ombudsman investigation.
"We will now carefully consider the inquest findings to help ensure such incidents are not repeated," he said.
Hampshire County Council said it would "carefully consider the full findings".
A mental health trust has pledged to improve after it was criticised at an inquest over "gross failures" before a prisoner killed himself in prison.