In general, the composition of two monadic adjunctions is not necessarily itself monadic. Two known counterexamples include $\mathbf{TorsionFreeAb} \to \mathbf{Ab} \to \mathbf{Set}$ and $\mathbf{Cat} \to \mathbf{Quiver} \to \mathbf{Set} \times \mathbf{Set}$ (the latter is because $\mathbf{Cat}$, the category of small categories, is not regular).
Now, let $C$ be a category with a monad $T$ and $S$ be a monad on the Kleisli category $C_T$. Then, for any two objects $X$ and $Y$ in $C$, morphisms from $X$ to $Y$ in the Kleisli category $(C_T)_S$ correspond to morphisms from $X$ to $SY$ in $C_T$, which in turn correspond to morphisms from $X$ to $TSY$ in $C$. This suggests that $(C_T)_S$ is also the Kleisli category of a monad $\bar{S}$ on $C$ for which $\bar{S}X = TSX$ on objects.
Question:
Does there always actually exist such a monad $\bar{S}$? In other words, is the composition of two Kleisli adjunctions always itself a Kleisli adjunction?