Mary Duncan was 17 when she disappeared from her home in Alexandria, West Dunbartonshire, in 1976.
The teenager, who had a 13-month-old daughter, vanished after telling her family she was going to meet a friend.
Her daughter later died of natural causes.
No trace of Mary has ever been found despite extensive inquiries.
Police are now beginning to search a number of sites at the Vale of Leven Hospital in Alexandria. The work started yesterday.
Detective Superintendent Calum Young, from West Dunbartonshire CID, said as Mary had not been heard of for more than 40 years, police must consider the option that she may have come to harm.
He said: “Mary left her home at 17 years of age, with no money and has never claimed benefits, paid tax or national insurance.
“She also left behind her baby girl Laura, who family say she absolutely adored.
“Specialist officers will be searching a number of locations at the Vale of Leven Hospital in connection with the investigation.”
The detective said it was still a functioning hospital and disruption must be kept to an absolute minimum.
He said: “It is also vital that the search team are left to get on with this important work and any updates will be provided in due course.”
Police said searches at two sites at the hospital will continue for a number of weeks.
Marie Farrell, director of Clyde sector at NHS Greater Glasgow and Clyde, said: “We have been working with Police Scotland on the planned excavation of a number of sites in the grounds of the Vale of Leven hospital.
“We would like to reassure patients that there will be no disruption to patient care.”
Officers have previously searched the home where Mary was listed as living on court papers from 2016.
Mary, who police have previously said they suspect had fallen pregnant again when she vanished, left her home in Third Avenue, Bonhill, on 19 March 1976 after telling her family she was going to meet a friend. She never returned.
Det Supt Young said Mary’s stepfather Norman Duncan, who was jailed for five years for historical sexual offences against girls, had been spoken to as one of a number of lines of inquiry.
Duncan, 73, was jailed in 2016 for abusing four girls during the 1960s and 1970s.
His victims were aged between eight and 15 years old when they were attacked between 1968 and 1976.
Mary’s sisters Debbie, Marion and Mandy issued a public appeal in 2018 in a bid to find out what happened to Mary.