A couple of things come to mind Dave.  With a car that old, grounds and
connections always become areas of suspicion.  Clean, tight, and solid
mechanically at the connector.  It's even plausible for faulty connections
to be heat related, which would be consistent with it running for an hour
before having a problem.

Did you have Sears test the system during a failure period?  That would be
appropriate.  I think, given the description, I'd initially be lean towards
an alternator, even though the diagnostics did not point that way.  When an
alternator is not working properly, the car will run off of the battery
until the battery voltage drops too low.  At that point, the world ends.
I'll bet it's not so much a matter of it cooling off as it is that it takes
time to recharge the battery by jumping it.

As for the voltage regulator - I don't know if the 98 Sonata has an external
regulator.  It's been a long time since I've seen a car with one, so I'd
kind of doubt it, but that's a guess.  Have you looked at www.hmaservice.com
yet?  It's a free website by Hyundai and it has service manuals for all
American Hyundai's.  You'll need to use IE (Mozilla and other browsers won't
work), and you'll need to download the Adobe SVG viewer - go to the site
requirements link, you'll see.  There's a world of information on that site,
although it's one of the worst sites that has ever been written, from a web
site development perspective.  Oh well - guess you can't have everything.
That site may answer the question of the voltage regulator.