According to Kelly's Trade Directory Zissman Brothers were at 334 Coventry Road in 1938, but by 1950 there was no number 344. I would take a guess that it was bombed out during the war. That's probably why your family moved to Great Barr. Then according to Kelly's in 1965 Moss Zissman opened a shop at 536 Coventry Road.
Interestingly neither location show back housing or terraces reachable from Coventry road at the rear of either shops according to OS maps of the locations on the dates given, is it possible that your family lived in the first floor of the shop premises accessed at the rear of the property.