Can Quick Search be set up so External ID does not have to be exact match to work?
I originally wrote it that way to avoid confusion. If you want to pull up contact id 20 you'll be overwhelmed with irrelevant results (201, 202, 203, 120, 220, 320, and so on). You'll get so many results that the one you actually wanted might not even be in the list!

That said, if someone had time and/or money for the project, I'm sure there could be something smart added to the query that made the exact match show up first in the list followed by partial matches.