Richard Lyon, 59, from Kirkmichael, South Ayrshire, has not been seen since around 7am on August 2.
Two days later he posted his car keys to the family home in an unfranked brown envelope, leaving his concerned family and friends to mount a search for his silver Honda CRV.
Police said they were unable to trace where the letter was posted because, while it had a stamp on it, it had not been franked at a post office.
They said they were aware that the self-employed Maybole-based land surveyor had been worried about his business recently and that this “may have been preying on his mind”.
His car was spotted on the A80 near Castlecary just before 1.30pm on the day he went missing, but police said it has not been seen since.
Mr Lyon’s son, David, who flew from his home in Australia for the public appeal, said his father, originally from Dundee, was a “private” man and was unlikely to ask for help.
He said: “He is very sociable but he is also quiet at the same time and he will always do something for someone else. He wouldn’t have been asking for help because he is very self-sufficient.
“This is a one-off. It’s totally out of character for him. He would always check in with my mum or my brother, let them know what he was doing. We are a very close-knit family.”
He said his father had walked most of Scotland’s Munros and tended to wear casual clothes --possibly deck shoes with casual jeans or khaki trousers.
Speaking at Ayr police station, he added: “Just to emphasise, when my dad would go walking, like most people, he would go to get away and have some fresh air and get peace of mind and relax. So we are thinking that’s hopefully where he’s gone and people need to jog their memories.
“He would have been wanting to look after the family’s interest first and foremost. He wouldn’t want a big fuss. He wouldn’t have foreseen this happening, having this sort of mass exposure.
“Please, please help us. This next few days are vital for finding the car and my dad.”
Mr Lyon’s wife of 32 years, Noreen, said the couple had been due to go on holiday to Yorkshire last Thursday.
“Anytime he went hillwalking he would go with somebody and give me a six-figure reference where he was going, where he parked the car,” she said. “I always wanted that. I wanted that information, so in a way I restricted his going to the hills to a certain extent.”
Strathclyde Police said Mr Lyon had been feeling “rather low” before he disappeared.
Chief Inspector Neil Kerr, of Ayr police station, said: “As time passes we are becoming increasingly concerned for the well-being of Mr Lyon.
“I am keen to speak to anyone who may have seen his vehicle, a Honda Civic CRV registration plate RJ55 EBP, as it may lead us to Mr Lyon’s location.
“I can confirm that Mr Lyon posted his car and house keys to the family home on Thursday, August 4, so I would appeal to anyone who may have seen, or who knows where, the car is currently located.”
Mr Lyon is 5ft 10in, of medium build and has grey receding hair and beard. He also wears glasses. When he was last seen by his family he was wearing jeans, a T-shirt and trainers.