My suggestion would be to add another AirPort Extreme, which will make the setup very easy and fast, and both AirPort Extremes will be displayed when you open AirPort Utility once things are set up. Online sellers like Amazon and Ebay are still offering Apple AirPorts for sale.
If you are thinking of another brand of router to add to the AirPort Extreme, we do not have a specific brand or model recommendation for you, since we handle questions about Apple routers on this forum.
Make sure that the router that you are considering can be configured to operate in Bridge Mode and act as a Wireless Access Point, also known as WAP. The WAP would be configured to provide a wireless network that uses the exact same wireless network name, same wireless security and same password as your AirPort Extreme.
The Ethernet cable will connect from one of the three LAN <--> Ethernet ports on the AirPort Extreme to the WAN port on the router that will be used to extend the network.