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.
On Saturday Greek police said they were fast-tracking toxicology reports in order to ascertain the cause of her death, potentially “in the coming days”.
Test results can take anywhere from several weeks to a few months depending on the complexity of the case and the workload of forensic laboratories.
They were also looking to trace Lisa’s American friends, who they said were also staying in Kypseli.
First Minister John Swinney called it an “absolutely appalling case”.