In the Wright Omega function, the following is used as the subscript for the Lambert W function:
$$\left\lceil\frac{\Im(z)-\pi}{2\pi}\right\rceil$$
Here is the full function for context (from the Wikipedia page):
$$\omega(z)=W_{\left\lceil\frac{\Im(z)-\pi}{2\pi}\right\rceil}e^z$$
I understand that $W_0$ is the real branch of the Lambert W function, and that $W_{-1}$ is the imaginary branch. But in this case, the subscript can end up being numbers other than $0$ and $-1$.
So what does this subscript mean in this case?
That's the "unwinding number",
$$\mathcal K(z)=\left\lceil\frac{\Im z-\pi}{2\pi}\right\rceil$$
which satisfies the relation
$$\log\exp z=z+2\pi i\mathcal K(z)$$
and is indeed an integer. If you read the linked paper, you'll notice that the definition there is a little different, but Corless and Jeffrey explained in their paper on Wright $\omega$ why they modified the definition.
This fits in perfectly fine with the Lambert function, which does expect integer subscripts in the usual case; however, if you read their paper further, you'll find that the Wright function then allows the extension of the Lambert function $W_k(z)$ to arbitrary $k$!
Anyway, read the two papers I linked to if you need more details.