Of course it's not a problem there are hotels open all year. Mid October would see virtually every one open. The summer season ends at the end of October, so somewhat less choice but that's all.
In mid October I would opt for Rhodes Town as there will still be lots going on there and you can easily visit other areas from that base. Many of the other resort areas will be very quiet at that time as it is coming to the end of the season.
Have a look at travelrepublic.co.uk for great cheap deals on hotels and accomodation.
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