Hi,
the easiest way - put search module inside menu as one of items:
https://www.gavick.com/documentation/jo ... nu-modules.
the hard way - put new module position instead, for example [login] if you do not use it, but it request code customization
OR
the 3th way - use "real" sidebar module which allows you to put on left or right side of your browser screen and put search icon and when somebody click -> hover a search bar. In JED search "sidebar" or "floating modules" or similar ...
It's right time to be big boy, and make decision. Have fun.