Refining Search Function Provided for HTM5 Output

I have an HTML Help system created by publishing my Framemaker book file to HTML5. When the Help is created, it has the Search function provided by Adobe. Our customers are finding that the search results are too general/broad. When they enter a term to search on, they often get back a long list of results that they have to wade through, making the Search function somewhat useless. 

I have discovered that if they enter multi-word terms surrounded by double quotes, you can target the search results a bit. Is there anything I can do within Framemaker (before I generate the HTML5) to improve or target the search results provided?

The development team for our product has suggested that they can provide search functions in the HTML5 files after they are generated, but we were hoping maybe there was something I could do within the source files to provide a better search function. 

Please note, I do have an index and that works fine in the Help, but users often just type in the Search box instead of using the index. Thanks in advance!