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

I am looking at seasonal jobs around Omaha that might lead to something longer. There are 12 people counting on me, so I am trying to be careful. I would like to avoid a gap in my résumé. Which employers treat seasonal workers well?

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

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.