@kunatir 

If you want everything to be one network, set the S4 into access point mode. This loses the router features of the S4, but you have your one network. You can leave the S4 in router mode if you want to keep the routing features though, and just disable the Wi-Fi on your Verizion. 

Here are the differences between the router and access point features: https://www.tp-link.com/us/support/faq/2399/

If the S4 is running in access point mode you can have all three wired to the router if you would like, but you at least need one wired. The rest of the nodes will be able to connect automatically to the main node via Wi-Fi. If the S4 is running in router mode, then you would only have the main node wired to the Verizon device, the rest would connect via Wi-Fi to the main node, or each other, or if you daisy chain them or have the S4 go to a switch, then the rest of the nodes wired to that switch.