  In this paper, we study several variants of Hahn-Banach smoothness, viz.,
property-$(SU)$/$(HB)$/$(wU)$, where property-$(SU)$ and property-$(HB)$ are
stronger notions and property-$(wU)$ is a weaker notion of Hahn-Banach
smoothness. We characterize property-$(wU)$ and property-$(HB)$. It is observed
that $L_1(\mu)$ has property-$(wU)$ in $L_1(\mu,(\mathbb{R}^2,\|.\|_2))$ but it
does not have property-$(U)$ in $L_1(\mu,(\mathbb{R}^2,\|.\|_2))$ for a
non-atomic measure $\mu$. We derive a sufficient condition when property-$(wU)$
is equivalent to property-$(U)$ of a subspace. It is observed that these
properties are separably determined. Finally, finite-dimensional and finite
co-dimensional subspaces of $c_0$, $\ell_p$ ($1\leq p<\infty$) having these
properties are characterized.
