Kos Town is very pretty but I would say the best time to visit if possible is the evening when it is much cooler. We have just returned from two weeks staying in Tingaki and Kos Town was only a 20 minute bus ride from us. We went into Kos Town late afternoon had a good look around and had a lovely meal there. We caught the 10 o'clock bus back to Tingaki.
We have just got back from Tingaki (stayed at the Mythos Apartments) this restaurant is just a couple of doors up from the Mythos. Although we tried several other places kept going back to it, the lamb kleftiko was fantastic! Could never remember the name but we always called it the 'Blue chair and check tablecloth place'. Very friendly, very traditional food. You won't be rushed, I don't think they would mind how long you stayed to be honest. I don't think we ever felt rushed in any of the restaurants in Tingaki.