Sorry, backup a little. I'd assumed the extra wire was one with a connector at the end, but now I'm wondering if you mean just a single wire hanging around?
If this is a small wire and there's an empty hole in the connector already plugged into the back of the alt...then get new connector and hook new wires into it...think that'd be best...you can't really just stick a wire into a connector hoping it'll stick...it'd be very squirrley!
If the wire is 1/4" thick or so hanging loose, that's the cable to battery...and it should be bolted under the rectangular black rubber cover on back of alt.
Sorry if this confuses you...does me too!