  We develop Stein's method for $\alpha$-stable approximation with
$\alpha\in(0,1]$, continuing the recent line of research by Xu \cite{lihu} and
Chen, Nourdin and Xu \cite{C-N-X} in the case $\alpha\in(1,2).$ The main
results include an intrinsic upper bound for the error of the approximation in
a variant of Wasserstein distance that involves the characterizing differential
operators for stable distributions, and an application to the generalized
central limit theorem. Due to the lack of first moment for the approximating
sequence in the latter result, we appeal to an additional truncation procedure
and investigate fine regularity properties of the solution to Stein's equation.
