Is it possible to chain Hall sensors?
The &quot;1-wire&quot; bus protocol would suit your requirements. Do a search and see if there's a &quot;1-wire Hall sensor&quot;. These can then be read directly by your microcontroller without any shift-registers.
1-wire devices are pre-programmed with unique addresses. An interesting little problem for you would be the registering the addresses of each sensor at each square. I suggest that you have a learning mode where you move a piece sequentially around the board, row by row, while recording which sensor is at which grid reference.