This is the steps to modify the List Views that display when you use the global search.
Decide on the maximum of 10 columns you want in your Search Layouts for each object
Add that list of columns to your documentation - you will refer to it often. (See Views - they should probably be the same).
Go to Search Layouts for the object you want to change
Click the Edit next to the Search Results
Add the fields you want, and put them in the right order, check Override the search result column customization for all users. click Save
The view is now updated.
Search for a Contact and you will see the new view.
Like Recent Views, Search Results, can NOT be modified to show different views to different users. Eg if you have a Partner Community and give them access to their own Accounts or Contacts, then they can see the same fields as internal users (unless of course the field has the security set correctly).