Agree that the GA flying in Cyprus is limiting. You generally have to stick to set routes and it won't take long to fly them all. On the upside, there are great opportunities to cross borders to the middle east, north africa and Italy.
You could try: Akrotiri Flying Club 00357 2527 6217. Its an RAF club so you may need to explain some kind of military connection. If you don't have any military connection its still worth giving them a try as they are much cheaper than the rest.