Client has router on one side of the house, resulting in poor Internet on the kitchen side. The cost-effective method was to set up an extender, which has been just fine in filling the kitchen with Wi-Fi. Now the home owner wants to extend Wi-Fi out in the backyard to get Sirius XM on their smart phone. Do I dare extend the extender? It would only be for low throughput music streaming. No web browsing needed. Is extending an extender absolutely the wrong thing to do?

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?

MoCA, Poweline either of those an option?

Extenders REALLY suck as they cut bandwidth in half, remember 802.11 is NOT a mesh network it is point to point.

Advise to do WAP. Ubiquiti WAP works great.

Wireless extenders should always be the last resort after all better options have been ruled out. I would treat the space the same as an indoor and get a WAP that covers the area without a lot of overflow. Mount outside and use something with an antenna pattern that fits like a 90 or 180 degree pattern as needed. Just make sure it's an outdoor rated WAP. POE would be advised.
FWIW, I spec in and install a lot of outdoor WAPs due to people taking smart devices outside.

After discovering A small smart panel and moving the modem to another room, there were enough cables to add a Ubiquiti access point that is now giving great coverage in all the key areas. Many thanks to all who offered excellent advice!