If you're looking to move into a larger apartment building, you can start looking now. The management company will generally know what apartments will be available for your move-in date even if they aren't listed online. You can have the whole process done 2 months before your actual move-in date.

If you're going for something operated by a private landlord or smaller leasing company, give yourself 2 months to start browsing and really get serious about it in late March. Personally, I'd aim to avoid unnecessary stress and have everything squared away in the first few days of April.