There is no such "nice" $C$ (other than $\Omega$ and $\varnothing$) in dimensions other than $1$.

Let $x_0 \in \Omega \cap \partial C \setminus$. If $h_\varphi$ exists, then $\varphi$ is equal to a real-analytic function $h_\varphi$ on $\partial C \cap B(x_0, \varepsilon)$. This is clearly not the case for the subharmonic function $\varphi(x) = |x - x_0|$ (given that the boundary of $C$ is sufficiently smooth near $x_0$).

There is no such "nice" $C$ (other than $\Omega$ and $\varnothing$) in dimensions other than $1$.

Suppose otherwise, that a non-empty, smooth, open set $C$ in $\mathcal{A}(\Omega)$ exists. If $C \ne \Omega$, then there is $x_0 \in \Omega \cap \partial C$. If $h_\varphi$ exists, then $\varphi$ is equal to a real-analytic function $h_\varphi$ on $\partial C \cap B(x_0, \varepsilon)$. This is clearly not the case for the subharmonic function $\varphi(x) = |x - x_0|$ (given that the boundary of $C$ is sufficiently smooth near $x_0$).