I 100% understand where you’re coming from and especially if you have a good relationship with your landlord it wouldn’t hurt to politely request a prorated refund of part of the last month’s rent. However, if they refuse it’s unlikely to be worth your time and/or money to pursue it further.

Since the landlord presumably didn’t pressure you to move out early and you turned in the keys it’s well within normative practice to look for the next tenant ASAP. And if you were under a lease it’s normative for them to expect full pay for your lease term even if you choose to move out early. The law might provide some specifics that would be in your favor beyond that but I have my doubts - it’s just one of the downsides of renting. Unless the landlord just decides it’s the nice or ethical thing to do you’re probably out that money.