Setting Up 3 Network Printers Connected To an Old Router That Is Connected To A Router / Modem Combo?

I am trying to setup 3 network printers for a client. He has an Xfinity modem / router, and there is only 1 free ethernet port available on the router. To avoid buying a switch, I used an old router he had lying around. I plugged in the 3 network printers into ports 2, 3, and 4. When plugging the WAN port from the old router to the empty port on the Xfinity modem / router none of the printers show up on the network. A single device does show up as a connected device in the Xfinity admin settings. When I remove the connection from the old routers WAN port and the empty ethernet port on the Xfinity modem, and plug that into port 1 on the old router, 1 of the network printers shows up. So the Xfinity router empty ethernet port is plugged into the old routers port 1. The 3 network printers are plugged into ports 2, 3, and 4, but only 1 of those printers shows up to add in Windows. It doesn't seem to matter which order I plug these in, still only 1 shows up. Is there a solution to get all of the network printers to show up?