As @Tyson already said in a comment, you don't need a router. To get more Ethernet ports, you only need a switch; and to extend WiFi network range, you only need a WiFi access point.
It just happens that the "home wireless routers" have a router, a switch, and a WiFi AP integrated in one device for convenience. But they're definitely not meant to be daisy-chained.
So, since you've already purchased the router, try running it in pure bridge mode, avoiding the 'routing' features and merely using it as an additional switch+AP.
Sometimes it's actually labelled "bridge mode", but sometimes you'll have to manually turn off DHCP and connect the router through its LAN port instead of WAN port.
Well, it is possible to make daisy-chaining work. But since that way you essentially create two separate networks, it means 1) it's a good thing that they use different IP ranges, 2) you must use different WiFi network names, which means you lose automatic roaming between the two.
So you would need to turn off NAT on the 'inner' router; then add a static route on the 'outer' router – for the 'inner' IP range and with the inner router as the route's gateway.
(That is, as long as your router allows it. I still cannot fathom the existence of "home routers" that lock away the ability to manually set routes, but such garbage actually exists.)