In your current setup it sounds like the cable modem is also a router. If your wired switch is the router, then everything I’m going to say is wrong. If the cable modem has more than one ethernet port, then it is almost definitely a router.
You have two options for the way things will work:
Cable modem/router —> Wireless/wired —> wired switch
Cable modem —> Wireless/wired/router —> wired switch
(each —> represents a physical cable.)
My suggestions is to use your existing router, and reconfigure the new piece of equipment to behave the way you want. Instead of using the uplink/WAN connection port on the wireless router, just use one of the internal/LAN ports to connect to the cable modem. This will allow the cable modem to continue acting like a router, doing NAT and handing out IP addresses. The wireless router will essentially become a wired switch, which happens to have a wireless base station, too. You might have to do some configuration on the wireless router so that it has an IP address that is accessible from the inside of your network. That’s not strictly necessary for the wireless router to function, but it will help when you want to connect to it and manage it. You will also want to disable the DHCP server on the wireless router. If you have two DHCP servers on the network, that will be bad.
Be sure and realize, my suggested method is creating a single internal network that spans both wired and wireless. If you care at all about security, be sure and enable WPA2 security on the wireless side. There are a variety of other arrangements you can do that would have wireless on its own network, which could be firewalled from your wired network, but that quickly gets complicated.