Michael Bryn Jones, 39, from Llandudno, disappeared on 3 April 2016 after going to the door of the Hergest psychiatric unit at Ysbyty Gwynedd, Bangor.
He was found hanging in woodland on 21 June.
Betsi Cadwaladr University Health Board (BCUHB) said it has already changed its procedures.
An inquest into Mr Jones' death heard he had been a patient at the Hergest unit until a few days before his disappearance and continued to suffer anxiety and paranoia after he left hospital.
He turned up at the unit in the early hours of 3 April and was sent to the accident and emergency department but left without booking in or speaking to anyone.
Robat Hughes, the senior nurse who spoke to Mr Jones at the unit, told the inquest he regretted not asking him whether he was already having treatment.
"He came to the door and asked to see a doctor, but I said there was no doctor there as she'd gone home sick.  I said that if he went to accident and emergency then there were doctors there," he said.
"He was relaxed and calm.  I didn't know if he was using mental health services and I didn't ask.
"I should have asked if he'd been having treatment.  It has been something I've thought about a lot since."
Dr Stuart Porter, a consultant psychiatrist who reviewed the incident, said: "Somebody should have taken Michael Bryn Jones to accident and emergency.  It's also good practice to follow that up with a phone call."
In a statement, BCUHB said it offered "sincere condolences" to Mr Jones' family and "fully accepted" the coroner's findings.
"We have carried out our own thorough investigation, in conjunction with Michael's family, and as a result we have made changes to service provision," it added.
Recording a conclusion of suicide, coroner Nicola Jones said: "There should have been more effort to persuade [Mr Jones] to come into the Hergest unit for a full assessment of his condition.
"He went to that unit looking for help, and he didn't get it."
A man who went missing after going to hospital with mental health problems should have been given more help, a coroner has said.