Seasonal jobs that might turn permanent - what worked for you?

A staffing agency says some roles convert to permanent after 7 weeks. I worked as a janitorial supervisor before this. I would like to avoid a gap in my résumé. Is it worth applying to several at once?

Thanks for writing this out. I went through something close to it 7 months ago, 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.