  The Freudenthal--Tits magic square $\mathfrak{m}(\mathbb{A}_1,\mathbb{A}_2)$
for $\mathbb{A}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}$ of semi-simple Lie
algebras can be extended by including the sextonions $\mathbb{S}$. A series of
non-reductive Lie algebras naturally appear in the new row associated with the
sextonions, which we will call the \textit{intermediate exceptional series},
with the largest one as the intermediate Lie algebra $E_{7+1/2}$ constructed by
Landsberg--Manivel. We study various aspects of the intermediate vertex
operator (super)algebras associated with the intermediate exceptional series,
including rationality, coset constructions, irreducible modules,
(super)characters and modular linear differential equations. For all
$\mathfrak{g}_I$ belonging to the intermediate exceptional series, the
intermediate VOA $L_1(\mathfrak{g}_I)$ has characters of irreducible modules
coinciding with those of the simple rational $C_2$-cofinite $W$-algebra
$W_{-h^\vee/6}(\mathfrak{g},f_\theta)$ studied by Kawasetsu, with $\mathfrak{g}
$ belonging to the Cvitanovi\'c--Deligne exceptional series. We propose some
new intermediate VOA $L_k(\mathfrak{g}_I)$ with integer level $k$ and
investigate their properties. For example, for the intermediate Lie algebra
$D_{6+1/2}$ between $D_6$ and $E_7$ in the subexceptional series and also in
Vogel's projective plane, we find that the intermediate VOA $L_2(D_{6+1/2})$
has a simple current extension to a SVOA with four irreducible Neveu--Schwarz
modules. We also provide some (super) coset constructions such as
$L_2(E_7)/L_2(D_{6+1/2})$ and $L_1(D_{6+1/2})^{\otimes2}\!/L_2(D_{6+1/2})$. In
the end, we find that the theta blocks associated with the intermediate
exceptional series produce some new holomorphic Jacobi forms of critical weight
and lattice index.
