Daniel McSwiggan, a 69-year-old with Alzheimer's disease, disappeared on Thursday September 12.
Police inquiries have now revealed he was last seen at Hamilton Low Park, South Lanarkshire, that evening.
Staff working at the starters office at the golf course and driving range within the park spoke to him at around 6.50pm. After that, he walked towards the first tee area.
Officers are urging any golfers, dog walkers or other passers by in the area at that time to contact them.
Inspector Alistair Anderson, of Wishaw police office, said: "I would specifically appeal to any golfers, dog walkers or anyone else who was in Hamilton Low Park or neighbouring areas on the evening of Thursday September 12 to contact me as soon as possible, as they may have information which can help in our inquiry to trace Mr McSwiggan."
He added: "I would urge anyone who has any information on this inquiry, no matter how insignificant it may seem, to contact Police Scotland officers on 101."
Following Mr McSwiggan's disappearance, specialist police units, including dogs, search teams and a helicopter have been involved in the search.
His family are said to be "completely distraught".
Mr McSwiggan is 5ft 7in and bald with grey hair at the sides and a moustache. When he was last seen he was wearing a stone-coloured jacket, dark grey trousers, a dark blue v-neck jumper, blue shirt and black shoes.