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."

The discount fashion chain said it had come to its attention that the footwear "does not meet the Primark usual high standards for chemical compliance".
The products in question are men's flip-flops in blue, black and khaki.
The company said customers will be offered a full refund and do not have to produce proof of purchase.
Primark, which is owned by Associated British Foods, said the footwear was sold in stores between 4 January and 2 June this year as part of its Cedar Wood State range.
"We have found levels of a restricted substance in the product in excess of the 1.0 mg/kg requirement," it said on the website.
A Primark spokesperson confirmed that the chemical in question was chrysene, used in dark coloured dyes, but said it was present at levels that would pose a minimal health and safety risk to customers.
The fault was discovered by Primark following up an inquiry by a third party, the company said.
"We take the safety of our customers, and the quality of our products very seriously," the spokesperson said.
The company has suspended all new orders from the factory that manufactures the flip-flops while the matter is investigated.
A man who went missing after going to hospital with mental health problems should have been given more help, a coroner has said.
[NEXT_CONCEPT]
Primark has recalled thousands of men's flip-flops over fears they may contain dangerous levels of a cancer-causing chemical.