  We prove a number of results related to the computational complexity of
recognizing well-covered graphs. Let $k$ and $s$ be positive integers and let
$G$ be a graph. Then $G$ is said
  - $\mathbf{W_k}$ if for any $k$ pairwise disjoint independent vertex sets
$A_1, \dots, A_k$ in $G$, there exist $k$ pairwise disjoint maximum independent
sets $S_1, \dots,S_k$ in $G$ such that $A_i \subseteq S_i$ for $i \in [k]$.
  - $\mathbf{E_s}$ if every independent set in $G$ of size at most $s$ is
contained in a maximum independent set in $G$.
  Chv\'atal and Slater (1993) and Sankaranarayana and Stewart (1992) famously
showed that recognizing $\mathbf{W_1}$ graphs or, equivalently, well-covered
graphs is coNP-complete. We extend this result by showing that recognizing
$\mathbf{W_{k+1}}$ graphs in either $\mathbf{W_k}$ or $\mathbf{E_s}$ graphs is
coNP-complete. This answers a question of Levit and Tankus (2023) and
strengthens a theorem of Feghali and Marin (2024). We also show that
recognizing $\mathbf{E_{s+1}}$ graphs is $\Theta_2^p$-complete even in
$\mathbf{E_s}$ graphs, where $\Theta_2^p = \text{P}^{\text{NP}[\log]}$ is the
class of problems solvable in polynomial time using a logarithmic number of
calls to a SAT oracle. This strengthens a theorem of Berg\'e, Busson, Feghali
and Watrigant (2023). We also obtain the complete picture of the complexity of
recognizing chordal $\mathbf{W_k}$ and $\mathbf{E_s}$ graphs which, in
particular, simplifies and generalizes a result of Dettlaff, Henning and Topp
(2023).
