After you installed the new SSD did you go into Disk Management and allocate and format the new drive? That drive has to be allocated or the system will not be aware it is there. This web page has more information. The procedure for Windows 11 is similar but you can also click on search in the taskbar and type in disk management.
I just got an xps8940 with 512 gb m.2 and want to add a second ssd in the 2.5" bay. I bought a Samsung 1 TB ssd. I installed it in the blue plastic holder, slid it in, plugged in the two cables that are pre placed at that location. When I boot up I can't find the new drive, which I think should be the E drive, as the disc drive is D:. It's been awhile since I have putzed with the internals. It has Windows 11 pro.
XPS 8940, adding Samsung 1TB SSD, not seen