Seasonal jobs that might turn permanent - any advice?

I am looking at seasonal jobs around Chicago that might lead to something longer. I have read a lot of threads here and could not find my exact situation. Hours ramp up in the coming weeks. Is it worth applying to several at once?

Thanks for writing this out. I went through something close to it last year, and it settled down once I kept a simple log of every call and form.

Keep a list of supervisors' names. A reference from one of them helped me land my next job.