You can find information on the internet that it is best to hike along the Lycian Way from April to May and from September to the end of October. It is because during summer it is too hot on the trail and during winter months it is rainy. Fair enough, but I was lucky when I hiked in January, there was just 1 rainy day out of 10.
I personally think that it is best to hike around the 15th of October - to the 20th of November and from the 1st of March - to the 15th of April. In March and November, it is about 20℃ in Turkey and you might think the season is over or not yet started. I want to warn you that on the trail, the temperature feels different! It is not windy there, so for me, this is the perfect time to hike. You might even want to swim in the sea! That is what we have actually done when we hiked in March and November numerous times.
You can find below a more detailed summary of my hiking experience at different times of the year.