When you use Ctrl+F in the document screen, the search term in the previous view is automatically set into the search box, so you don't have to retype it.
Another approach would be to add a mechanism to copy search terms from the view to the clipboard when searching. The reason is that the search term is available not only in the document but also in a file attached to the document by Ctrl+V on it's search box.
If you are searching with more than one word, I think one idea would be to select the first word. Do you have a better idea?