Daniel McSwiggan, a 69 year-old with Alzheimer's disease, disappeared on Thursday, September 12.
Police inquiries have now revealed that he was last seen at Hamilton Low Park, South Lanarkshire, that evening.
This was revealed as officers launched a separate search for another vulnerable man who went missing on Tuesday.
James Coogans, 30, of Coatbridge was last seen in the town's Main Street.
The search for Mr McSwiggan has been boosted by the news that staff working at Hamilton Low Park golf course and driving range spoke to him at around 6.50pm on the night he went missing.
After that, he walked towards the first tee area. Police 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: "Following new information which we received from previous appeals, specialist police search teams and local officers will be in the Hamilton Low Park area today as part of our investigations to locate Mr McSwiggan."
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.
Officers searching for James Coogans have also urged anyone who may have seen him to get in touch.
He is 5ft 5 ins, with short brown hair and of slim build. He has blue eyes and a scar on the right side of his face.
When last seen he was wearing a grey hooded top with a front pocket, blue jeans with white writing down the sides and black shoes.
Police and James' family are said to be "concerned" about his safety. Anyone with information should call police on 101.