It doesn't have to be this way. Unfortunately, we live in a throw-away society where we feel it's alright to get rid of something once it becomes an inconvenience. Don't be quick to jump on the first apartment you see. There'll probably be a better one available soon. Widen your search.
Most people only look as far as the classifieds ads.
Many landlords list their property through real estate agents or rental associations rather than the classifieds. Take advantage of rental services that help tenants find apartment. Ask friends, relatives and co-workers to keep an eye open for you. Many apartments are rented via word of mouth before they're ever advertised in the papers. A home that allows pets might be in a different neighborhood than you'd prefer.