Yeah, OR is tricky.
The only thing I can think to do here is create a master "concatenated" field of all your field inputs.
Then give the user a quick filter on wildcard (you could expose multi-select if you want, but that could be really ugly).
That way, they can type a full or partial entry and get the results.
You could duplicate the concatenated field to give them a "2nd wildcard" filter to further refine the result set.
(and repeat duplication for however many you think would be useful)