It follows a pilot project run by NHS 24 with NHS Forth Valley, which led to a 27% reduction in patient no-shows.
After receiving written confirmation of their appointment, patients are telephoned eight days before the appointment to remind them to attend.
Results showed that the number of missed appointments in outpatient paediatrics, which had become as high as a quarter, fell to one in six.
Across the specialities as a whole, the rate reduction for the pilot in missed appointments was 27%.
A similar patient reminder service was started at the beginning of May with NHS Western Isles and is currently operating across all outpatient appointments.
Several other health boards have expressed an interest in the service.
Launching the extended programme in Glasgow, Health Secretary Nicola Sturgeon said: "It is every patients' responsibility to keep their appointments, but we also recognise that the NHS has a role to play in helping to make appointments convenient for patients.
"The reminders have already achieved great results and with more health boards set to trial them we can hope to see a better service for patients and a reduction in NHS time wasted."