I don't think any kind of home search can be started too early. Depending upon the apartment complex or property management company you talk to, they all should be able to forecast to a certain extent whether something will be able and may even be able to have you go ahead and reserve something. The rental market is hot right now so it's better to give yourself more time.
Good luck with your move! If I can do anything for you in the future, feel free to contact me!