When is the best time to start applying for Fall employment?

I’ll be graduating from my undergrad in June, and I already have a summer job lined up. However, I’m worried about making sure that I have a full-time job by the Fall (starting late August / early September). When would be the best time to start applying for jobs that start in Fall? Should I do so in the summer or keep looking now? It seems that most of the job openings I find need me to start in June.

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