Seasonal jobs that might turn permanent - 12 months and counting

A staffing agency says some roles convert to permanent after 5 weeks. I would rather hear the honest version than the polite one. 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 12 months ago, and it got easier once I kept a simple log of every call and form.