The Lycian Way is doable from October or so, if you are not too heat averse. The northern two-thirds of the St Paul's Trail are ok from sometime in September, due to the higher altitude.
However, if you decide to hike in Turkey, I would suggest going to the Kackars in the northeast of the country. There's no set trail, but you can put together your own walk, varying the length and difficulty to suit yourselves. Kate Clow (who devised the Lycian Way and SPT) has published a book of walks, available on Amazon. The route descriptions are about the same quality as the books for the other 2 walks, but the print quality, map quality and GPS waymarks (available from the website) are better.
July, August and September are the best months for the Kackars. They are very beautiful and water is much more available (in fact, rain is guaranteed on the northern side).