  We explore the $k$-smoothness of bounded linear operators between Banach
spaces, using the newly introduced notion of index of smoothness. The
characterization of the $k$-smoothness of operators between Hilbert spaces
follows as a direct consequence of our study. We also investigate the
$k$-smoothness of operators between polyhedral Banach spaces. In particular, we
show that the $k$-smoothness of rank $1$ operators between polyhedral spaces
depends heavily on the dimension of the corresponding spaces rather than the
geometry of the spaces. The results obtained in this article generalize and
improve upon the existing results in the $k$-smoothness of operators between
Banach spaces.
