Seasonal jobs that might turn permanent (update)

I am looking at seasonal jobs around Cleveland that might lead to something longer. There are 6 people counting on me, so I am trying to be careful. I would like to avoid a gap in my résumé. Is it worth applying to several at once?

Apply to several. Start dates and schedules change quickly and having options helps.

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