Proper wiring sequence

If there is more than one device, there needs to be a router immediately after the modem. No way around that. You can have 2 routers in a network. You just need to configure one of them so that the DHCP range is not the same as the other one.

Well the question here is, can the modem do the DHCP/NAT for the network or is it a bridging modem where it is expecting a broadband router behind it?

*IF* the modem itself can act as your DHCP/NAT device then you can simply disable the DHCP functionality of your wireless AP and connect it to the network via one of its network ports (not the WAN port).

If the modem acts as a bridge and you need a broadband router to connect multiple devices then you would need to either get another broadband router OR run two cables to the wireless AP. One cable for the WAN to modem connection and one that connects the wireless AP to your switch.

If you can't move the switches as well (too costly to rerun any existing lines) than I would just run 2 cat lines to the new wirless location (unless your modem is doing the DHCP/NAT like mym6 says). If you run 2 lines, you could run from the modem to the router and back down to the switch allowing you to keep your original setup of modem - router - switch - devices.

Please note that in the case where mym6 says the modem handles your DHCP and NAT, it is a router/modem. Not all modems will do this and are made to handle 1 device at a time. Some ISPs are releasing modems with routers built in and some with even wireless, so it depends on your modem.

For ease of sake, I would go with 2 cat6 cables to your Router's new location. Another option is to just run 1 wire and put another Access point (with the same SSID, your equipment should move along them without issue choosing the one with the most signal), or just buy a wireless extender (Cisco makes one that my mom uses with great success). It just plugs into a wall and picks up the signal and repeats any data sent to it (very simple explanation of what it does). Same thing as the AP, it uses the same SSID so that your equipment just chooses the one with the strongest signal.