Make a name for yourself by doing something other people value. You'll acquire contacts naturally that may eventually lead to a job. Search the various online job databases, especially the ones that focus on nonprofits. Check out Goodworks, the Chronicle of Higher Education, Opportunity NOCS, Nonprofit Jobs. Go to individual web sites for organizations that especially interest you.
Many places post jobs on their web sites.
Monitor the newsgroups and email lists that often carry job opening announcements. One such newsgroup is soc.org.nonprofits. Check out the guides for nonprofits mentioned in the Heliport section for email list references. Participate in newsgroup and list discussions, forming friendships and professional relationships with peers in the field. Ask around informally to learn what's available.