  The covering number of a non-linear character $\chi$ of a finite group $G$ is
the least positive integer $k$ such that every irreducible character of $G$
occurs in $\chi^k$. We determine the covering numbers of irreducible characters
of the symmetric group $S_n$ indexed by certain two-row partitions (and their
conjugates), namely $(n-2,2)$ and $((n+1)/2, (n-1)/2)$ when $n$ is odd. We also
determine the covering numbers of irreducible characters indexed by certain
hook-partitions (and their conjugates), namely $(n-2,1^2)$, the almost
self-conjugate hooks $(n/2+1, 1^{n/2-1})$ when $n$ is even, and the
self-conjugate hooks $((n+1)/2, 1^{(n-1)/2})$ when $n$ is odd.
