The world-first scheme, aimed at tackling period poverty, was successfully piloted in Aberdeen.
The Scottish Government is to provide charity FareShare with more than £500,000 to extend it to reach an estimated 18,800 more people.
The organisation will use its centres in Aberdeen, Dundee, Glasgow and Edinburgh to begin distributing products over the summer.
An evaluation of the Aberdeen pilot showed that two thirds of the 1,082 women and girls who took part had experienced difficulties in accessing sanitary products in the past.
Equalities Secretary Angela Constance said: “It is unacceptable that anyone in Scotland should be unable to access sanitary products and I am pleased that we are able to work with FareShare to make products available more widely through the services delivered by their partners.”
Head of FareShare in Scotland Gillian Kynoch said: “FareShare works with over 1,000 community organisations and charities across Scotland.
“We are excited to be working with the Scottish Government to use this network to make sanitary products available to people across Scotland.”
Free sanitary products will also be available to those at school, college or university from August.
Labour MSP Monica Lennon welcomed the extension of the scheme, but called for a statutory requirement to ensure free provision in schools, colleges and universities as well as “placing a duty on the Scottish Government to deliver a free universal system of access”.
“Scotland can be a world leader in tackling period poverty if we are bold enough to take these radical steps,” she said.