Could you specify which 'search box' are you using? Is that minimized list box or a special extension made by third party? Usually, if a list box is placed close to edge of window, it will adapt its popup directions and positions itself.
However, for a full screen selection, you might try 'search tool' button on right upper corner and see does that solve you problem:
You might want to check this video out to see the various ways you can work with selections.
Please let us know how you do.
Please mark the appropriate replies as helpful / correct so our team and other members know that your question(s) has been answered to your satisfaction.