  It is known that there is no extremal singly even self-dual $[n,n/2,d]$ code
with minimal shadow for $(n,d)=(24m+2,4m+4)$, $(24m+4,4m+4)$, $(24m+6,4m+4)$,
$(24m+10,4m+4)$ and $(24m+22,4m+6)$. In this paper, we study singly even
self-dual codes with minimal shadow having minimum weight $d-2$ for these
$(n,d)$. For $n=24m+2$, $24m+4$ and $24m+10$, we show that the weight
enumerator of a singly even self-dual $[n,n/2,4m+2]$ code with minimal shadow
is uniquely determined and we also show that there is no singly even self-dual
$[n,n/2,4m+2]$ code with minimal shadow for $m \ge 155$, $m \ge 156$ and $m \ge
160$, respectively. We demonstrate that the weight enumerator of a singly even
self-dual code with minimal shadow is not uniquely determined for parameters
$[24m+6,12m+3,4m+2]$ and $[24m+22,12m+11,4m+4]$.
