Margaret Fleming, 36, was last seen at her home in Inverkip by her two carers at about 17:40 on Friday 28 October.
She is described as about 5ft 5in tall, with collar-length black hair, brown eyes and has a heavy build, although her weight can fluctuate.
Police had said they were trying to build a picture of Ms Fleming's life, part of which she kept "quite private".
When last seen, she was wearing a green tartan fleece or jumper, dark jeans or trousers, and dark Karrimor-type boots. She also had a satchel-type handbag.
A police spokesman said: "There is a specialist search team combing the area around where the missing person was last seen, this includes in the garden of her last known address."
Speaking earlier this month, Ch Insp Elliot Brown, area commander for Greenock, said: "We are currently working with limited information and whilst it does seem like this is quite an unusual missing person case, it is challenging.
"As we continue to try to piece together Margaret's life, we understand she kept a side of her life quite private so we're trying to look into this in a bit more detail to see if this will help with our inquiries.
"We are working backwards to try to find out more about Margaret, so we've been speaking to friends and family members to establish a clearer picture of her life over the years."
The detective said that Ms Fleming was a student at James Watt College in Greenock between 1996 and 1997.
He said he was keen to speak to anyone who remembered her from then, and who might have been in touch with her over the years.
Police have searched the home and garden of a vulnerable woman who has been missing for a month in Inverclyde.