With the recent changes at Google, fixing the search functionality for .gov websites becomes more closely tied to the user experience. This means bringing in user experience professionals who have deep knowledge/expertise in search technology (that powers the site search for these sites) OR search professionals who has a strong background in user experience and information architecture. All too often, designing a successful search experience is either general information architect (and often without their knowledge or consent) or to the developers (again, without explicit knowledge of this responsibility or enough background in the field to be successful).
It should all start with a strategy that incorporates how the "client" perceives their service/offering integrated with how customers search for this service/offering. The outcome is a core list of term phrases that represents this intersection and becomes a tool for benchmarking success, spotting content gaps, designing user assistance tools and planning enhancements to the site.
Successful search is achieved by design build on an explicit strategy that is conceived with the help of professionals with specific expertise.