  We consider the split special orthogonal group $\mathrm{SO}_{N}$ defined over
a $p$-adic field. We determine the structure of any $L$-packet of
$\mathrm{SO}_{N}$ containing a simple supercuspidal representation (in the
sense of Gross--Reeder). We also determine its endoscopic lift to a general
linear group. Combined with the explicit local Langlands correspondence for
simple supercuspidal representations of general linear groups, this leads us to
get an explicit description of the $L$-parameter as a representation of the
Weil group of $F$. Our result is new when $p=2$ and our method provides a new
proof even when $p\neq2$.
