Connect the secondary router to one of the LAN ports of the Verizon supplied router (probably the Actiontec MI-424WR). Change the base address of the linksys to something that doesn't conflict with the Actiontec, I suggest 192.168.2.1 instead of it's default.
That should allow connections. If you need to port forward for the computer, you'll have to do some fancier configuration.
Of course, the way I'd consider is to connect the secondary router as a wireless switch, which will allow you to use the 802.11n link, and not add the complication of two routers. Here's the recipe for doing that.
Connecting two (or more) SOHO broadband routers together.
Note: The "primary" router can be an actual router, a software gateway like Microsoft Internet Connection Sharing, or a server connection that has the capability to supply more than one IP address using DHCP server capability. No changes are made to the primary "router" configuration.
Configure the IP address of the secondary router(s) to be in the same subnet as the primary router, but out of the range of the DHCP server in the primary router. For instance DHCP server addresses 192.168.0.2 through 192.168.0.100, I'd assign the secondary router 192.168.0.254 as it's IP address, 192.168.0.253 for another router, etc.
Note: Do this first, as you will have to reboot the computer to connect to the router again for the remaining changes.
Disable the DHCP server in the secondary router.
Setup the wireless section just the way you would if it was the primary router, channels, encryption, etc.
Connect from the primary router's LAN port to one of the LAN ports on the secondary router. If there is no uplink port and neither of the routers have auto-sensing ports, use a cross-over cable. Leave the WAN port unconnected!
This procedure bypasses the routing function (NAT layer) and configures the router as a switch (or wireless access point for wireless routers).