**Notation:** Let $x$ be a point of a smooth separated finite type DM stack $\mathcal X$ over a field. Suppose

• $G$ is the stabilizer of $x$,

• $V$ is the tangent space of $x$ (which comes equipped with an action of $G$),

• $G^\textrm{triv}\subseteq G$ is the subgroup which acts trivially on $V$,

• $H = G/G^\textrm{triv}$ (note $H$ acts on $V$),

• $K$ is the subgroup of $H$ generated by pseudoreflections on $V$, and

• $K'$ the commutator subgroup of $K$.

$\mathcal X$ can be expressed as you described (in an étale neighborhood of $x$) if and only if $K'$ is trivial.

I'll now unpack that answer. Any smooth separated finite type DM stack over a field can be (canonically!) obtained from its coarse space with the following steps (this is basically the main Theorem of my paper with Matt Satriano, A "bottom up" characterization of smooth Deligne-Mumford stacks):

- take the canonical stack of the coarse space,
- do a root stack construction along the ramification divisor of the coarse space map, rooting each component of the ramification divisor by the degree of ramification,
- take the canonical stack again (the root stack may not be smooth any more, but it will have quotient singularities!), and
- add a gerbe.

Your question is "when can we skip step 3?" That is, when is the root stack from step 2 already smooth?

Using the notation above, and looking formally locally around $x$ (so we can assume $\mathcal X=[V/G]$; you can describe it étale locally too, but it's clearer this way), the above steps are:

- $\bigl[(V/K)/(H/K)\bigr]$ is the canonical stack of the coarse space $V/H = V/G$,
- $\bigl[(V/K')/(H/K')\bigr]$ is a root stack of $\bigl[(V/K)/(H/K)\bigr]$,
- $[V/H]$ is the canonical stack of $\bigl[(V/K')/(H/K')\bigr]$, and
- $\mathcal X = [V/G]$ is a $G^\textrm{triv}$-gerbe over $[V/H]$.

Note that step 1 is the familiar way of building the canonical stack of a space with quotient singularities (in this case $V/H$) by expressing it as a quotient by a finite group *somehow*, and then quotienting out the subgroup generated by pseudoreflections, with the Chevalley-Shephard-Todd theorem ensuring that you don't lose smoothness. This description tells us that the canonical stack is any description of the space as a quotient where the group acts without pseudoreflections.

Note that in step 3, we're quotienting out by $K'$, which has no pseudoreflections since it's a commutator subgroup (all its elements must therefore act with determinant 1, and there are no pseudoreflections of determinant 1). So it makes sense that this step is a canonical stack.

It's pretty clear that step 4 is a (trivial) gerbe.

Seeing that step 2 is a root stack is more complicated; you can find the details in the section titled "A local description of Theorem 1" in the paper linked above.