Briefly, my question is about whether there exists conservative extensions of $T_0\colon=\mathbf{ZFC}$ other than $T_1\colon=\mathbf{NBG}$, specifically extensions of $\mathbf{NBG}$ obtained in a way similar to the extension of $\mathbf{ZFC}$ to $\mathbf{NBG}$, which will be clarified shortly. I believe it best to leave our variables unsorted, for clarity and to avoid cumbersome notation.
To clarify my remark regarding the means of extension, identifying languages with their set of formulas, $\mathbf{NBG}$ is an extension in the expansion of the usual language $\mathcal{L}_0=\mathrm{L}(\epsilon)$ of set theory (identifying languages with their set of formulas) to the language $\mathcal{L}_1=\mathrm{L}(\epsilon,\mu_0)$ and among the axioms of $\mathbf{NBG}$ is the sentence $\forall x(\varphi_0(x)\leftrightarrow\exists y(x\in y))$. Hence, I am referring to a similar approach by adding to $\mathcal{L}_0$ unary relation symbols $\mu_\alpha$ for each $\alpha<\beta$ and some ordinal $\beta$ while adding axioms similar to the previous one.
It goes without saying in moving from $T_0$ to $T_1$, for every $\mathcal{L}_0$-sentence $\sigma$ if its relativisation to $\mu_0(v)$ is $\sigma^{\mu_0(v)}$ ($v$ not occurring in $\sigma$ of course) then we want (and do have) \begin{equation} T_0\models\sigma \quad\iff\quad T_1\models\sigma^{\mu_0(v)} \end{equation}
To add some terminology for convenience, (also) refer to sets as $0$-classes, classes as $1$-classes and $\alpha$-classes for objects belonging to the domain of discourse of the theory $T_\alpha$, up to the ordinal $\alpha$ fo which this is possible or otherwise for every ordinal $\alpha$. Is this at least possible for all $\alpha<\omega$ or even $\alpha=\omega$? In case it is not clear, I would like that \begin{equation} T_\alpha\models\forall x(\mu_{\gamma}(x)\rightarrow \mu_{\delta}(x)),\quad\text{whenever }\gamma<\delta<\alpha. \end{equation}
Yes, we can do this at least through the computable ordinals (so well beyond $\omega$). One way to do this is described here; basically, the idea is to have level $\alpha$ look like $L_\alpha(M)$ for some $M\models ZFC$ (analogously to how the conservativity of $NBG$ over $ZFC$ is to show that $L_1(M)$ is a model of $NBG$ whenever $M\models ZFC$.
(Note that here I'm using $L_\eta$ to denote the $\eta$th level of the constructible hierarchy, not as you've used it.)