There is a way to render a standalone search box on a search interface: https://developers.coveo.com/display/public/JsSearchV1/Adding+a+Standalone+Search+Box
But it overrides the redirect behavior and becomes simply an extansion of the search interface.
If you want your search box to redirect to another page, it won't work.
|I need it...||Quarter|
Fixed! Will be available in the July 2016 build