You may consider going to Chios and then to Cesme.
Here is more information: There are many different options from Piraeus to Chios. Prices start from 22.50€. From Chios to Cesme, I have found one line for 25€. This site has the information related to pricing and time tables. You need to select your departure and arrival points from the menus.
There is a ferry between Piraeus and Samos at about 50€, then another ferry between Samos and Kuşadası at about 35€.
Credit:stackoverflow.com‘