Apple engineers did their best to design the AirPort "extend" feature as a proprietary option that would only work with other Apple routers. They pretty much met their goals.
Through the years, we have not found another 3rd party "extender" or "repeater" that was fully compatible with the main AirPort router,. That doesn't mean that another 3rd party extender won't work.....it's just that using Linksys, Netgear, TP-Link, etc products, we have not found one that works correctly.
Your best bet would likely be to check Ebay for used or refurbished Apple AirPorts similar to your main router. That way, you know that the product will do exactly what you want.
Ideally, you would use the same model of router to extend the network, but an AirPort Express would do in a pinch if you don't need the highest attainable speeds or strongest signal in the extended area.