Eric, David (below) is right. You need to find a place that is not operated by a rental agency but rather by the owner. You also should get the current management to give you a letter showing your on time payment history. (If the only reason they are making you move was a past eviction.) Try and also show them your on time payment history with other purchases, perhaps a car or other bill.
You can also offer to pay the future owner one or two months up front, so you will always be ahead on your payments. Or offer to pay an additional months deposit. Something that will make them feel comfortable in taking a chance on you. You will always get this money back if you stick with the good payment history and you don't do any damage to the property.