We study the typical behavior of the harmonic measure in large critical Galton-Watson trees whose offspring distribution is in the domain of attraction of a stable distribution with index $α\in (1,2]$. Let $μ_n$ denote the hitting distribution of height $n$ by simple random walk on the critical Galton-Watson tree conditioned on non-extinction at generation $n$. We extend the results of arxiv:1502.05584 to prove that, with high probability, the mass of the harmonic measure $μ_n$ carried by a random vertex uniformly chosen from height $n$ is approximately equal to $n^{-λ_α}$, where the constant $λ_α>\frac{1}{α-1}$ depends only on the index $α$. In the analogous continuous model, this constant $λ_α$ turns out to be the typical local dimension of the continuous harmonic measure. Using an explicit formula for $λ_α$, we are able to show that $λ_α$ decreases with respect to $α\in(1,2]$, and it goes to infinity at the same speed as $(α-1)^{-2}$ when $α$ approaches 1.