I would start looking now.
Also, just because your lease is up, you may still have the option to either renew your lease for another term or continue on a month to month basis with the same terms that are still in place.
Just because your lease term expires, it does not mean that you have to move. And, depending on what type of property you are renting and where it is located, you may be covered under Rent Control.
Let me know if I may be of assistance. I offer a FREE property search to renters and buyers.
All the best,