Finding an apartment in Greece for renting may not be a big problem. There are many places that would be suitable for you depending on your budget and requirements. But the pertinent question for you will be where do you start? You can take the help of the receptionist of the hotel where you are staying or you can take the help of some broker for the purpose.
There may be questions about your family and the co-habitants in the apartment house that you wish to rent in places like Patras. This is natural for a conservative society as the Greeks. You will be surprised if you ask for an apartment to stay alone and get a response that none is available due to the high season there. It simply means that if they feel anything weird about you, they are not going to rent you an apartment in Greece. People in Greek Dodekanisa are conservative.
Another option for you is to look at the classifieds in the local newspapers and get hold of the advertisements on letting out traditional houses and apartments coming up there. The quoted rates will invariably be very high for your comfort and taste. The good news is that the owners can be negotiated and the rates in Greece are always negotiable. Once again, you may engage an agent who will take his charges but can find out the same apartment for you at a much less price.