A supported ethernet adapter should work.

Come to that, I think the machine I pulled the hard drive out of (and which was old enough it had Windoze98 on the other drive! 0O) might have been using a D-Link pci card, for which I still have the box it came in... and which I can probably pull and slap into the HP Compaq. If so, that WAS the card the distro was recognizing and using before. That machine is out in the ice-cold covered-patio right now, I'm not gonna tackle that till daylight again.

Yes, that reads like the most pragmatic option.

Install something modern. Don't try to use something that has not been updated for 9 years. There is no profit in it. No updates available, all the security risks.

https://www.centos.org/download/

You sound like you need to read this first.
https://wiki.centos.org/
https://docs.centos.org/en-US/docs/

Okay, moved the PCI Ethernet card over from the old machine, turned it on and... I have networking!

Also, oddly, it looks like the older machine also had an Ethernet port on the motherboard. I guess maybe I ran into the same issue on that machine that I did on the current one: it wouldn't "see" the native Ethernet port, but I set up the netpliancy thingy on that one so long ago that I've forgotten all details on it.

As for updating to a newer CentOS... that IS in the cards, but its one of these things were you're expected to install a whole new major-version-number update to the netpliancy system over the old, from a live-CD, but first I would need to do a full backup of the emails initiated from the web-interface of the old netpliancy system, just in case the update frells up the emails or something. (This is an email-machine netpliance thingy, btw.) My problem is, due to severe RAM limits on the old machine, I couldn't get the backup to launch, among other problems. 0o