The body of Dean Morrison was discovered at Barry Burn in Carnoustie at approximately 12.40pm on Sunday.
Although officers say the death is not being treated as suspicious, a witness appeal has been issued to try and piece together his last moments.
As well as urging anyone who saw the 40-year-old before his death, officers are now calling for help in their search for a scarf he was wearing when he was last seen on Saturday night.
Senior investigating officer, Detective Inspector Ray Birnie, said: “We are treating his death as unexplained and at this stage there is nothing to indicate it is suspicious.
“We need to find out what happened to Dean so that we can provide his family with the answers they both need and deserve.
“We urgently need to speak to anyone who may have seen Dean between 11.12pm on Saturday 15 December and 12.40pm on Sunday 16 December.
“If you think you may have seen him, perhaps walking in Carnoustie or within a licensed premises, then please get in touch.”
He added: “Dean was wearing a dark grey puffer jacket, scarf, dark shirt, black trousers and brown shoes.
“We are searching for the scarf which we know he was wearing when he was last seen. The scarf is made by Next and is multi-coloured striped including brown, green and silver.
“If anyone finds a scarf matching this description in Carnoustie, please leave it where it is and call us.”
Carnoustie residents are being told to expect increased police activity in the days ahead “including search officers, uniformed officers and the use of a drone”.
Detectives believe Mr Morrison caught a taxi from Broughty Ferry, near Dundee, on Saturday night to the Angus town.
Renewing his call for the driver to come forwards, Mr Birnie said: “Despite widespread media coverage and previous police appeals, we are again appealing for the taxi driver to contact us.
“It is absolutely vital that we find this driver. We know the driver picked Dean up at 11.13pm on Saturday 15 December in Fort Street/Brook Street in Broughty Ferry and may later have dropped him off in Carnoustie.
“The taxi itself was a black people carrier style with a sliding rear door and an orange light on the roof.
“We urgently need to speak to the taxi driver to find out where he dropped Dean off.”
Police Scotland said Mr Morrison had 16 years of service covering the Tayside area.
Chief Superintendent Andrew Todd, Divisional Commander for Tayside Division, added: “Dean was a highly respected officer, who was extremely well liked and popular with his colleagues. He was thoroughly professional in his work.
“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.”
Anyone with any information is asked contact Police Scotland on 101 quoting 1960, or speak to any police officers.
Alternatively people with information can contact the independent charity Crimestoppers anonymously on 0800 555 111 or via their anonymous online form at crimestoppers-uk.org.