Oh okay, I see.
In that case I suggest rather creating another search instance as well (you can duplicate the first one for ease of use) to have some separation. On the header search remove the settings as you don’t need that.
Then place the second search bar into a sidebar position on the results page. Check the theme builder/editor if it offers any sidebar widgets, usually modern themes allow adding another column where you can put any shortcode or widget and place the second search shortcode there.
This way you will have the header search separately configured and they will not conflict.