Informal authority means the type of authority that does not depend on formal positions. You are "authorized" by people who like you or who see value in what you do.
In a nutshell, expanding your "informal authority" means:
Networking
- "Selling" your idea in small chunks.
- Connecting with stakeholders
- Test driving some small portions of the project in order to "score early wins" (Heifetz et al., p. 135). For instance, if you are about to launch a major change in responsibilities, you may pick a small group of allies and ask them to try the change first.