Suggested device to extend Wi-Fi but keep same SSID?

Our ISP is BT Broadband and we use their BT Smart Hub (sometimes called the Home Hub 6), which was released in 2016.

The router (only just) struggles to reach every part of the home.

We had a similar issue at a different house seven years ago. We had signal in a specific room but it was very poor, so we got a device that extended the router's Wi-Fi signal. However, the device created a separate SSID and, since the connection to the router was very poor but not zero, we had to keep manually connecting to the new SSID when we were in this specific room as no phone would automatically switch, then connect back to the router's SSID when we left the room. It was very frustrating.

Is there a better solution, these days?

We can not replace the router.

We are hoping to buy a device that extends the signal (we don't need to extend it by much - so maybe a £50 upper limit?) but that (hopefully) retains the SSID so that phones do not stick to the barely-working signal.

I work for BT as an "end point engineer", where part of my job is to fix this kind of issue for people, so here's my professional advice.

Ring BT and ask them what their price would be for them to provide you with a BT Smart Hub 2 and a single BT WiFi disc. It'll do exactly what you want it to do and it'll do it very well, and you may even get it for free if you ask nicely enough.

The ubiquiti stuff is really good but you'd ideally need to cable to the access point. You can mesh them but you'd need 2 access points.

The BT pricing model is / was the discs as a service, but they can be bought on eBay relatively cheap, especially compared to the rental charges accumulated over a couple of years. However, I set up an Asus wireless AX router in "AP Mode" where it acts as a wireless access point for the existing wifi network. PM me if you want some help setting up such a device.

TPLink Deco's what you want, or something similar like Ero etc.