Yes, you can do it a couple ways, but watch out you do not want 2 DHCP services running on the same network.
One: Linksys WAN port run it to the Verizon router. This will give you two different subnets, along with if you are on the verizon router you will not be able to get to a PC on the linksys router unless you port forward and set static IP's.
Two: in the Linksys router disable the DHCP service and run a Ethernet cable from one port (on the linksys) to a port on the Verizon router. Then you will only have one subnet and the PC's will be able to see each other. Your DHCP server will be on the verizon router.
I personally use the 1st one for security reasons, and have all my PC's sitting behind the 2nd router... Main reason is that the ISP can see everything on the first network, however due to network protocols and standards, they can not see past the 2nd router.
Take note that I have nothing to hide, but its none of their business what I have running on MY home network. In my experience in the network field, some people abuse certain privileges, and I will not be a victim of an admin with a god complex.
Use what ever set up suits you best, and happy networking