@david_ten Did the posts / responses answer your questions?
Yes, I believe so for now.
Thanks again to all for the help.
Based on the responses here and my sleuthing around on the internet, it seems a good start would be to purchase a switch and try it in different positions either near my router or near my Antipodes DX server. I also plan to power my router, and possibly the switch too (if it needs power), using my HD Plex linear power supply that is currently not being used.
As far as which switch, it seems a
Cisco WS-C2960G-8TC-L Catalyst 2960 8-Port 10/100/1000 Ethernet Switch is pretty well-regarded and would be a good start. It can be purchased for less than $100 used or less than $200 "refurbished." Should I assume I need 10/100/1000 ports? The port configuration (speed?) seems to make a difference in the price, and they are not all the same.
Another "audiophile" switch I came across in my searching that is not mentioned in the OP is the Silent Angel Bonn N8 Switch for about $400 new, from Crux.