How hard is it to rent a house or apartment with a poor rental history?

My wife and I were renting a tiny two bedroom apartment, paying our rent on time, but once our baby was born we needed more space. We asked our landlord to let us move into a bigger unit, but they refused so we left to live in a bigger place before our lease was up. Then, while staying in the larger place, we both lost our jobs and again had to vacate before our lease was up.Now we need a place to stay and have already been turned down twice and of course we know why. What im wondering is, do we have any chance of being able to rent agian?If so, how? when? any suggestions from anyone who may have been in a similar situation in the past?