Lisa had initially stayed in the area of Keratsini, 5.9 miles (9.5km) west of central Athens, with Greek friends.
She had told them that on 15 July she would go to Kypseli, where her “American friends” or acquaintances were.
Police sources said officers were looking to identify these individuals.
On Thursday, a spokesperson for Greek police told the BBC they were awaiting the outcome of the toxicology tests to determine the cause of death and ascertain whether foul play was involved.
A post-mortem examination is under way, with police saying they cannot yet confirm whether her death is considered murder.
On Wednesday, the Foreign, Commonwealth and Development Office (FCDO) confirmed they were providing consular support and were in contact with the Greek authorities.
First Minister John Swinney called it an “absolutely appalling case”, and said her family would be supported by the Scottish government and the FCDO.
