Airport products are still available on eBay. Airport tends to work best with "its own kind", I have had a few hiccups using a mixed setup. Airports do not work really well as wireless extenders (they split the radio for client and uplink tasks, cutting bandwidth in half at least. But, using Ethernet to link them, you can get decent performance from the second usit.
Powerline adapters like
this can get Ethernet to the center of the dead zones. These are sold at Best Buy, Fry's, etc so if Amazon delays delivery, look elsewhere. The idea is, plug one into the Extreme and then one into the second WiFi device in the remote area. Don't plug into power strips, must be directly plugged into the wall. It should give you over 1Gbps over the power lines.
Then, get an Airport Express like
this. If you have a lot of devices, or need faster uplinks, a used AP Extreme like
this in the back area would give you 1Gbps uplinks over power lines. Remember, your ISP service may be the slowest link in the network, so if you have <100Mbps, the Express with its 100Mbps ethernet may be enough. But if your ISP > 100Mbps, go with the Extreme as the second device to get max speed for wireless devices in the back of the house.
Using Airport Utility, setup the second Airport in Bridged mode (network tab), and then setup WiFi like you did on the router. A few tips, you can use a different network name (SSID) for each device if you want to make sure devices connect to one or the other, if there is overlap, they may pick a WiFi device that has intermittent performance and cling to it. But with different SSID, you can setup devices to only connect to the one that gives consistent good results in the area. iPhones, iPads, etc will easily switch between networks if there is a clear advantage. Also, when possible, use Wired connections to leave more bandwidth available to wireless devices. Finally, like with different SSID for the two routers, different SSID for 2.4 and 5Ghz can help prioritize which signal devices connect to.
If you are not tied to keeping your current router, there are some good Mesh solutions. I use Synology RT2600AC + RT2200, the signal in the rear of the home would need to be strong enough to reach the second device unless you want to use power line ethernet to connect them. Other Mesh systems include Netgear Orbi, Eero, TP Link Deco... I am not as familiar with these, but in your situation, wirelessly connecting the second (or third) mesh access points might be a problem if the signal is that weak in the back of the house, so make sure they can work with Ethernet uplinks to the router.