  Let $F$ be a probability measure on $\mathbb{R}$ in the domain of attraction
of a stable law with exponent $\alpha\in (0, 1)$. We establish integral
criteria on $F$ that significantly expand the probabilistic approach to Strong
Renewal Theorems (SRTs). The criterion for $\alpha \in (0,1/2]$ is much weaker
than currently available ones and in some cases provides sufficient and
necessary conditions for the SRT. The criterion for $\alpha \in (1/2, 1)$
establishes the SRT in full generality and in a unified way, barring the Limit
Local Theorems employed. As an application, for infinitely divisible $F$, an
integral criterion on its L\'evy measure is established for the SRT. As another
application, for $F$ in the domain of attraction of a stable law without
centering, an integral criterion on $F$ is established for the SRT for the
ladder height process of a random walk with step distribution $F$.
