The rental market is a very tough and competitive market right now. I just leased out a really nice unit in Cypress for $1800/month. http://www.9708paseodeoro.epropertysites.com/. This one is in a really really nice area and I had 8 applications. Working with an agent will be really helpful to get you into places quickly and they would submit your application on your behalf. The other bonus is it doesnt cost you a penny, the seller pays the agent. We use the MLS for properties available for lease, I can set you up with a listingbook account which is directly linked to the MLS but you can totally control it. Another thing I can do it set you up on a drip so as soon as a property in your area, in your price comes available you would get an email. There isnt much, but you probably would be seeing a property or two a week that way and they would be current available properties. Good luck!
The MLS unfortunately only has a portion of available rentals. There isn't one site for all rentals, like the MLS for homes for sale. Working with a few local property managers can be helpful. Also, the old school task of driving around looking for "For Rent" signs with the owners number on it.