  In \cite{Kawasetsu:2018irs}, Kawasetsu proved that the simple W-algebra
associated with a minimal nilpotent element $W_{k}(\mathfrak{g},f_\theta)$ is
rational and $C_2$-cofinite for $\mathfrak{g}=D_4,E_6,E_7,E_8$ with
non-admissible level $k=-h^\vee/6$. In this paper, we study
${W}_{k}(\mathfrak{g},f_\theta)$ algebra for $\mathfrak{g}=E_6,E_7,E_8$ with
non-admissible level $k=-h^\vee/6+1$. We determine all irreducible (Ramond
twisted) modules, compute their characters and find coset constructions and
Hecke operator interpretations. These W-algebras are closely related to
intermediate Lie algebras and intermediate vertex subalgebras.
