The problem is, many tenants lie. They sign a year long lease because it’s cheaper, or more available, intending to break it. They really don’t care if the landlord has to pay extra for move in-move out fees, rental agencies, advertising, cleaning, empty properties. It’s not their problem if they can get away with breaking it, and especially if they can manage to keep their deposit too. Hey, that deposit was for the last month’s rent, right? We will rent to someone who appears honest with mediocre credit, far sooner than someone who is lying, with perfect credit.
Totally different situation if the tenant has been in there for a long time and now they have decided to look for a house, though not legally different. New tenants—they probably planned on this in the first place. Increased rent for just a month isn’t going to cover anything. What would you have charged if it was originally a six month lease? Probably much more, or you wouldn’t have rented to them. People take advantage, and I don’t know why you should suffer financial losses because of it. I’m sure they are just fine with any loss you have to take. What does the rental agent say?
Certainly, people can ask for anything they want. They can ask for five years free rent, just so you have the honor of renting to them. But you can consider their request, and say no. If you feel that you want to agree to the month to month, you could resign with them, with an agreement that they pay all costs associated with the disruption, rent for the time it’s empty, all fees. But it may be very difficult collecting on that.

