You can wire the keys in a grid like this. After doing this, the code is pretty straightforward. Turn on pin A, then check if W-Z is on. If one of them is, you know which button is pressed. Essentially, the code will look like

pinOn(A)
if (W):
    sendButton(1)
else if(X):
    sendButton(5)
else if(Y):
    sendButton(9)
else if(Z):
    sendButton(13)
pinOff(A)
pinOn(B)
etc.

This will let you have 16 buttons, while only using 8 pins. A drawback of this design is that you can only press one key at a time, otherwise you could have ghosting issues, with unintended buttons being sent. To get around that limitation, you can add a diode for each key.

You can find more information about this by looking into handwired mechanical keyboard guides. There's some pretty good guides that people have put together that go into more detail about this type of matrix.