What is the best way to improve WiFi coverage throughout a large home?

This depends on which specific WiFi router is currently installed.

Until about 2016, the only way to extend WiFi coverage was to connect WiFi "extenders" to the primary WiFi router.  The main drawback was that each extender created its own WiFi network.  When a mobile device moved around the house, it would remain connected to the WiFi access point until performance became so bad that the device elected to drop the connection and start looking for a better connection.  "Honey.  Why is the WiFi so SLOW???  Oh, you need to turn WiFi off and turn it back on so it will connect to a better node."

Mesh WiFi systems were created specifically to address this problem.  Taking advantage of some IEEE 802.11 standards, the device and WiFi system could communicate and facilitate a transition from one WiFi access point to another without the user noticing anything.  Every manufacturer of WiFi products soon created a mesh system (Asus, Amazon eero, Google, Linksys, Netgear, Ubiquiti, etc....)

The landscape is a bit more convoluted now because companies have released sort of 'hybrid' WiFi systems where WiFi "extenders" operate more like "mesh satellites" when connected to a router of the same brand.

So, the bottom line is

• First look at the current WiFi router and see if the company offers WiFi extenders which will create a seamless "mesh".  If not....
• Look at Mesh WiFi offerings and decide which most closely meets your situation in terms of cost, features, performance, etc.
  do you need support for WiFi7? (own any devices which might take advantage of it?)
  what is the speed of your internet connection? (do you need higher than gigabit connections?)