Let $X$ be a Banach space with dual space $X^{\ast}$, unit sphere $S(X)$ and unit ball $U(X)$. For every $x\in S(X)$ we set $\Sigma(x)=\{f\in S(X^{\ast})\colon f(x)=1\}$ and $$ M(X^{\ast})=\bigcup\limits_{x\in S(X)}\Sigma(x) $$ (the set of norm-attaining functionals). \par According to C. X. Nan\ and J. H. Wang\ [Chinese Ann. Math. Ser. A {\bf 11} (1990), no.~3, 321--324; MR1072625 (91i:46016)], a point $x\in S(X)$ is called $k$-strongly smooth (in short, $k$-SS) for $U(X)$, if $\dim\Sigma(x)\leq k$ and if every sequence $\{f_{n}\}_{n\geq1}\subset S(X^{\ast})$ satisfying $f_{n}(x)\rightarrow1$ is relatively compact. In particular, the notion of a (Fréchet) smooth point corresponds to the case $k=0$, in which one has $\Sigma(x)=\{f\}$ and $f$ is strongly exposed by $x$ in $U(X^{\ast})$ (in particular $f$ is both an extreme and a $\sigma(X^{\ast},X)$-point of continuity for the dual ball). \par In this paper the author shows that a point $x\in S(X)$ is $k$-SS for $U(X)$ if and only if $J(x)$ is a $k$-SS for $U(X^{\ast\ast})$, where $J$ is the canonical embedding of $X$ in its second dual $X^{\ast\ast}$. Consequently, every $k$-SS point satisfies $$ \dim\Sigma(J(x))\leq k $$ (this latter property is hereby called $k$-very smoothness). \par Furthermore, calling $k$-extreme any point $x\in S(X)$ that cannot be represented as a nontrivial convex combination of $(k+1)$ linearly independent points of $S(X)$, another interesting result reads as follows: a space $X$ is $k$-SS (meaning that every $x\in S(X)$ is $k$-SS) if and only if every $f\in M(X^{\ast})$ is both a $k$-extreme and a $\sigma(X^{\ast},X)$-point of continuity of $U(X^{\ast})$. Finally, if $X$ is a locally $k$-uniformly rotund space [cf. F. Sullivan, Canad. J. Math. {\bf 31} (1979), no.~3, 628--636; MR0536368 (80h:46023)], then every $f\in M(X^{\ast})$ is a $k$-SS point of $U(X^{\ast})$.