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There are thousands of small not for profits across the country. Some are local, grassroots organizations, and some are larger membership/trade associations. There is actually a field called "Association Management", which can provide you with a variety of different types of positions to specialize in, or have a goal of becoming an Executive Director/CEO.
Entry level positions might be office assistant, membership coordinator, events coordinator, etc. Check the Nonprofit Times website for job listings, or Opportunity Knocks as well.