I'm not your lawyer. Iowa Code 562A.29(3) might be worth a look here. If you abandoned the unit (more on that later), and the new tenancy begins in a month that you have already paid in full, you should receive back the prorated portion of the rent that the new tenant is responsible for. The landlord doesn't get to double dip. So if you paid for March in full on 3/1, because your lease originally encompassed all of March, and then a new tenant moved in on 3/12, you are responsible for 11/31 (35.48%) of the rent and the landlord should write you a check for 20/31 (64.52%) of it.
It still might be worth talking to an attorney if you did not legally abandon the unit. "Abandonment" is a legal term and even if you told them that you were gonna move out, it is possible you may not have checked the legal boxes for this to be triggered. If you are paying the rent each month on time in accordance with your lease, the course of conduct between you and the landlord could be crucial to determine if you actually "abandoned" and gave rise to the landlord having the right to re-lease the property to someone else. If the landlord did not handle this properly, they might now be leasing the same property to two different people (including you) and that is what is called a legal no-no.
The easiest thing is probably to fire off a response requesting an $X refund based on the math above, maybe drop a cite to that Code chapter, and if that doesn't work then talk to an attorney. But again this is not legal advice and I'm not your lawyer.