PC Dean Morrison was found dead in Barry Burn in the Angus town on Sunday afternoon.
It is understood he had been returning home from a Christmas night out in nearby Broughty Ferry when he fell into the water beside the championship golf course.
PC Morrison was found dead by a passer-by the following morning. His death is not being treated as suspicious by police who said “all indications to date are that it was a tragic accident”.
Police said they had traced now a taxi driver they were looking for and confirmed he had been dropped off in Carnoustie around 11.30pm on Saturday night.
Welsh-born Dean, who is survived by wife Emily, 41, and his ten-year-old son, was then seen shortly after in the town’s Main Street.
They are now urging anyone who saw him after that to come forward.
Detective Inspector Ray Birnie, of Police Scotland’s Tayside Division, said: “Whilst our enquiry is still ongoing, I can confirm that Dean’s death is not being treated as suspicious. I would like to stress that all indications to date are that it was a tragic accident.
“Our thoughts remain firmly with his family, friends and colleagues at this very difficult time. We are of course continuing to support his family with specialist police officers.
“I can confirm that we have this afternoon spoken to the taxi driver who drove Dean to Carnoustie. He has confirmed that he dropped Dean off at the roundabout at Barry Road/MacDonald Smith Drive at 11.30pm on Saturday 15 December.
“A member of the public has also contacted us to tell us that Dean was seen walking in Main Street shortly thereafter.
“We still need to establish if anyone else has seen or heard Dean after this.
“As a reminder, Dean was wearing a dark grey ‘puffer’ style jacket, scarf, dark shirt, black trousers and brown shoes. We are still searching for the scarf which he was wearing. The scarf is from Next and is multi coloured striped, including brown, green and silver. If you find this scarf, please leave it where it is and phone us.
“There will be a continued increased police presence in Carnoustie whilst our enquiries are continuing. I would like to sincerely thank the public for their support and assistance. The help we have received to date is allowing us to gradually build a picture around Dean’s last moments.”
Police earlier paid tribute to PC Morrison, who had served for Tayside Police and Police Scotland for a total of 16 years in Dundee.
Dean was a highly respected officer, who was extremely well liked and popular with his colleagues. He was thoroughly professional in his work.
DI Birnie added: “His death is a great tragedy and he will be sadly missed by all who knew him and we will do all we can to support them during this distressing time.
“Our thoughts are with his family and friends.
“We ask that you respect their privacy at this difficult time.”