Least non-linear ordinary degree of a symmetric group

For $n\ge 3$, the least ordinary irreducible dimension exceeding one is $n-1$, except at $n=4$, where $S^{(2,2)}$ has dimension two. The dimension-$n-1$ shapes are $(n-1,1)$ and its conjugate partition, with the additional shapes $(3,3)$ and $(2,2,2)$ at $n=6$. For $n=2$ there is no non-linear irreducible. A proof uses the restriction branching rule for a symmetric group: a nonstandard shape with at least two corners has two non-linear predecessors, and a rectangular shape has one predecessor which in turn has two non-linear predecessors. This bounds its degree strictly above $n-1$ once $n\ge 9$; the smaller cases follow directly from the hook-length formula.