Although you may occasionally be faced with a tenant who will not leave as agreed, you will certainly encounter a tenant who leaves before the expiration of his/her lease and does not want to pay the balance of his/her financial obligation to you.  In these cases, in practically all states, you don’t have the right to demand that the tenant pay the rent due each month for the balance of the lease.

Confirm that the tenant has indeed given up his right of possession.  Then take reasonable steps to mitigate or limit the ongoing rent and other charges to the departed tenant by preparing the rental unit for re-renting.  In most states, you must make a reasonable effort to promptly re-rent the property, including advertising or using other usual rental marketing methods, or the tenant may be released from any further legal obligation to pay the balance of rent owed under the lease.

Handle the marketing of this rental unit just like any other unit.  You do not need to lower the rent, lower your tenant selection standards, or give this rental priority over any other available rental unit.  Of course, you cannot be vindictive and attempt to enrich yourself either.