We were at our house in Gumbet from 22 October to early November and the weather was lovely. We sat out on the balcony for breakfast every morning and walked round in sleeveless t-shirts and shorts every day. In the evening, we wore a light cardigan if we felt chilly.
We love it at the end of the season. Everybody has more time for you and you can walk round without hordes of others bumping into you (that sounds awful but I'm sure you know what I mean). Yes some of the shops and restaurants in Gumbet were closed and more closed at the very end of October but Bodrum was still alive - and much nicer than in the height of the season when you're likely to get "run over" by one of the "crocodiles of people" off the cruise ships!!
We have a pool at our place and although it was chilly, we found the sea okay to paddle in. What we liked also was that the heat was less fierce than in the summer so you could actually walk along the beach or around town without melting.
The peninsula is great to tour round as well and it's so easy by dolmus - and fun. We hire a car when we're over there and did some fantastic trips out to places where the dolmus don't go. Also, if you hire a car at that time of the year, the roads are quieter.
Hope you have a great time wherever you go.