Extending a vocal note

Really, Melodyne is the best solution for this. Indeed, you wouldn't have to splash out that much, as Melodyne Uno would do the job fine.
I am not a massive fan of Logic's pitch correction on vocals, I find that you can hear it. That said, YMMV. As for extending the note, you could try cutting and copying a part of the sustain and X-fade it, but I'm really not sure how well it would work...
You could also do this in the audio editor using the pitch and time machine. It really is a matter of trial and error, and beware that it is destructive. There's always undo, but do keep it it mind.

If you want to extend the note quickly in logic, just find a steady portion of it, cut it at the beginning and the end where the waveform crosses zero and insert this extra fragment in. I find that most of the time this will work just fine with a couple tries.

You should also try Logic's time stretching algorithm. If you're not stretching it too far, this may sound fine too. You can do that by separating the note into its own region, double clicking on it, choosing Factory: Time and Pitch Machine. Try all the different types of algorithms. Some sound way better than others on cetain types of material.