I read that the friends' house in Symi town is coastal and also a report that there walk was X miles but there were shortcuts. Here is the island - pedi on the right. If the villa was near the coffee place at the top of the photo he could have cut across some of the open terrain on the right of the photo.
Heatstroke, a fall, a fall into a well, a snake bite could have befallen him while off the beaten track. There also appear to be a few ruins of windmills and houses close by where he could have sought shelter if injured or unwell.
Perhaps yesterday the focus was on the coastal part of the walk and the search has lost time now they know he completed that part of the walk.