Extenders

The short answer is yes.  But rather than an extender, get a proper wifi system that can span the entire footprint of the home.  Disable the radios in the gateway and configure the personal router as as an access point for the simplest configuration.   

Got tp link deco because of $$ and reviews. Works well, no need for restarts or anything else.

Is that for replacing your main router and an extender? 

Does that replace the main router and add extenders?

@Hollywasahoodrat  An option, as mentioned above, is to purchase a separate router with a satellite(s) depending on the area you need to cover. You can either set up the new router/satellite as an access point as mentioned above or you can set it up as a mesh WiFi system (recommended for covering large areas). The gateway still acts as a modem to communicate with AT&T but the new router is either used for just extending your WiFi or it takes on the routing and WiFi duties (mesh). And, if you can hardwire the satellite(s) back to the router, that results in a more robust and reliable WiFi. Netgear, Eero, Ubiquiti, Google, etc all make good mesh WiFi systems depending on your needs.

"Off the shelf" extenders can work but you will probably be disappointed in the coverage or reliability.