From the answer to this question we know a space can be sequentially discrete but fail to be $k_2$-Hausdorff: Let $\beta\omega$ be the Stone-Čech compactification of the naturals, and let $X=\beta\omega\cup \{p'\}$, where $p'$ is a duplicate of some non-isolated point $p\in \beta\omega$, i.e., the neighborhoods of $p'$ are those of the form $\{p'\}\cup U\backslash \{p\}$ for some neighborhood $U\subseteq \beta\omega$ of $p$.
Then $X$ is sequentially discrete, yet not $k_2H$, since the map $f\colon \beta\omega\to X\times X$ given by
$$f(x)=\begin{cases}
(x, x) & x\neq p\\
(p,p') & x= p
\end{cases}
$$ witnesses that the diagonal is not $k_2$-closed.
While I believe $k_2H$ is the only property in $\pi$-base that is an immediate strengthening of $US$ with no other properties between it, we could also ask if sequentially discrete might imply $P$ for some other strengthening $P$ of $T_1$ that lies between $T_1$ and $T_2$. Then we would arguably have $US+P$ as a better strenthenening than just $US$.
However, the only other such strengthenings that are immediate strengthenings of $T_1$ (i.e., no other properties sandwiched between them) that I saw on $\pi$-base were $RC$ (retracts are closed), semi-Hausdorff, and locally Hausdorff.
This same example shows neither of the first two of these need be true: to see that $RC$ fails we observe that
$$r(x)=\begin{cases}
x & x\neq p'\\
p & x= p'
\end{cases}
$$
is a retraction, so that $\beta\omega\subset X$ is a non-closed retract.
Semi-Hausdorff fails, since $p$ and $p'$ cannot be separated by a regular open neighborhood of $p$.
$X$ does happen to be locally Hausdorff, but the cocountable topology on an uncountable set is sequentially discrete and not locally Hausdorff.
(Actually, on reflection it's easy to see that the cocountable topology fails to be semi-Hausdorff or $RC$ as well, but I will leave the other example as it stands, since the cocountable topology is $k_2$-Hausdorff, and much more.)
Remark.
On a more positive note, we do have (as remarked in the same answer linked above), that a sequentially discrete space will be "$SUS$" ("strongly" $US$ in that question), meaning continuous transfinite sequences have unique limits, as defined in more detail in the linked question.
