I imagine that if you are interested in dates after 1921 then it might be difficult — electoral registers as mentioned in reply #3?
The image below is Smithfield in White's Directory of 1919. This only gives details of businesses, but it indicates that Smithfield was organised into courts, so an address like "4 Smithfield" will relate to several households. Here are links to two large scale maps from 1889 (Smithfield is split across them) to show where the individual courts were – perhaps you already have this information, but it will flesh out this thread for any future readers.
OS Town Plans 1889
[Covered entrances or alleyways to courts are denoted by crossed lines.]
—lower section of Smithfield
https://maps.nls.uk/view/231282486#zoom=4.8&lat=9996&lon=13294&layers=BTCourts 1, 3, 5, 7, 9, 11, 13 (part) all on the west side below Cross Smithfield.
Courts 2, 4, 6 (part) on the east side.
—upper section of Smithfield
https://maps.nls.uk/view/231282471#zoom=5.3&lat=1786&lon=12744&layers=BTCourt 13 (part including entrance) on the west side.
Courts 6 (part) and 8 on the east side. Court 6 looks to be particularly complicated.
Going back to the first photo linked by MollyC in reply #7, the notes say that it is taken from near the entrance to Court 5, and that Court 9 "on the left has been demolished." Incidentally I discovered that if you click on Zoom above the image you get a higher resolution image (the basis of the 'magnify' function.