Seasonal jobs that might turn permanent after 5 years as a bookkeeper

A staffing agency says some roles convert to permanent after 5 weeks. I have been trying to stay organised about it. 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 8 months ago, and it settled down once I kept a simple log of every call and form.