The vast majority of landlords are not willing to rent for one month. A vacant rental home has just experienced the departure of a tenant, and the landlord has incurred costs - in time and money - to clean up, renovate, and prepare the property to be rented again. These investments are best recouped over at least a several month period. If you find a home you like but the landlord is unwilling to rent for one month, you might consider offering the landlord a higher payment for the single month to defray certain costs.
Otherwise, you might inquire at local hotels/motels and apartment complexes and inquire about 1-month rent.