If I understand correctly, extending the extender would cut network speed to a bit less than 1/4 what it is directly to the router. Can't you get ANY wires anywhere near the back yard? If not under a slab, maybe in an attic by a vent on the back yard side?
A good answer is easier with a clear question giving the make and model of everything. "The biggest problem in communication is the illusion that it has taken place." -- G. “Bernie” Shaw