The internet is a wonderful thing, but the rental information on the net is often wrong, out of date and/or otherwise not correct.
The rental market moves so fast that information can be out of date the day it is posted. Newspapers are also stale quickly.
Call a local professional in your market. Many agents and agencies specialize in rentals, seek them out. I am a rental specialist in my market and find it difficult to keep current on the inventory; on Mondays I assume everything available on Friday has been leased during the weekend. I confirm every appointment to make sure it is still available.
In addition to good information, you need the services of a real estate pro to actually get a lease signed. It is very common now to see multiple tenants trying to lease the same properties. It can get very ugly in these cases and literally there are very few rules. First come first served IS NOT THE RULE! It is the job of the property manager to to find the BEST tenant for their client the landlord.
A real estate profession knows how to get you the best possible chance at landing a scarce rental property. In many markets the landlord pays the tenants agent's fee.