Two routers and DD-WRT router

http://www.tomshardware.com/forum/
TC time capsule
Fios router
FIOS router connected to the TC: LAN port on Fios to WAN port on TC
I currently have a Fios network with the Verizon router's wireless radio disabled, Time Capsule running off a LAN port for wireless and wired to a server computer. I'd like to replace the wired connection to the server with a DD-WRT router. I imagine the way to do that would be to run the DD-WRT router off a different LAN port from the Fios router to the server and still leave the Time Capsule in the router for wireless purposes. 
Current set-up
192.168.1.1 (fios) and 192.168.1.2 (TC)
Fios Router (with wireless turned off) is the gateway. Running from the LAN port on there is a time capsule that I provides wireless to the house and ethernet to a desktop computer in the office.
Set up I would like
Fios router is the gateway. Time Capsule still serves wireless network for house. New router as a new network connects by ethernet to the desktop computer in the office (with the wireless on this router disabled).
Where would the New router go into the network? I imagine of the LAN port of the Fios router, but when I do that, I do not get internet connectivity through there (it does broadcast the wireless, but I can't access the router from the desktop)?

I'd like to run the dd-wrt router as gigabit-ethernet into the desktop (as a vpn) and keep the time capsule serving the wireless (under a different IP). So would it be better to run both the routers separately off the main Fios router (through LAN ports) or run the dd-wrt router behind the Time Capsule (from the TC's lan port)?
Whenever you chain routers together, LAN to WAN, you *must* use different networks, or else routing becomes ambiguous.
So if the Fios router is using the 192.168.1.x network, perhaps the TC should be using the 192.168.2.x network, or 10.0.0.x, or whatever, just as long as it's different. If you update the TC w/ a static IP of 192.168.2.1 and reboot it, it should automatically reconfigure the entire router w/ the 192.168.2.x network, including its DHCP server.
Here's the Fios

Here's the TC

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All I can really tell from those images is that indeed the two routers are using the same 192.168.1.x network. I would change the TC's static IP to 192.168.2.1, reboot it, and reboot the connected devices so they get IPs in the 192.168.2.x network.