How to add a button to the header area?

[Arkshine]

You can add that CSS to make the logo to stack above the search.

.home-logo-wrapper-outlet {
    z-index: 1;
}

You have another solution to CSS floating-search-input instead to avoid dealing with the sides (make sure to remove CSS on floating-search-input-wrapper) :

.floating-search-input {
    position: absolute;
    left: 0;
    transform: translateX(calc(50vw - (33rem / 2)));
}