Are there structural alternatives to Cardano’s radical formula for general cubic equations?

It is a classical result that the roots of a general cubic polynomial $x^3 + ax^2 + bx + c = 0$ can be expressed via Cardano’s formula using radicals of the form: $$x=\sqrt[3]u+\sqrt[3]v+k$$

where $u$ and $v$ depend on the coefficients and the discriminant $\Delta$. ​I am curious about the structural uniqueness of this radical representation. Specifically, I would like to know whether alternative algebraic forms involving different radical towers exist. ​For instance, is it possible to express the roots using a different nested radical structure, such as:

$$x=\sqrt{\sqrt[3]n+i\sqrt m}+k$$

or other combinations of radicals that do not reduce to the standard sum of two cube roots?

My understanding & motivation: From Galois theory, the splitting field of a general cubic over $\def\Q{\mathbb Q}\Q$ has a Galois group isomorphic to $S_3$, corresponding to a field extension of degree $6$. The standard radical tower corresponds to: $$\Q\subseteq\Q(\sqrt\Delta)\subseteq\Q(\sqrt\Delta)$$

Adding an outer square root like $\sqrt{\sqrt[3]{\dots}}$ would seem to force the degree of the radical extension tower to $12$, which exceeds $[K:\Q] = 6$.

Questions: ​Does Galois theory strictly mandate that any radical expression for the general cubic must reduce structurally to the Cardano-type tower (quadratic extension followed by a cubic extension)? ​Are there any non-standard radical representations (or alternative algebraic methods) for cubic equations that differ structurally from Cardano’s solution?