Let $S=(S_n)$ be an oscillatory random walk on the integer lattice $\mathbb{Z}$ with i.i.d. increments. Let $V_{\rm d}(x)$ be the renewal function of the strictly descending ladder height process for $S$. We obtain several sufficient conditions -- given in terms of the distribution function of the increment $S_1-S_0$ -- so that as $R\to\infty$ $$ (*) \quad P [ S\; \mbox{leaves $[0,R]$ on its upper side}\, |\, S_0=x] \, \sim\, V_{\rm d}(x)/V_{\rm d}(R)$$ uniformly for $0\leq x\leq R$. When $S$ is attracted to a stable process of index $0<α\leq 2$ and there exists $ρ= \lim P[S_n>0]$, the sufficient condition obtained are also necessary for $(*)$ and fulfilled if and only if $(α\vee 1)ρ=1$, and some asymptotic estimates of the probability on the left side of $(*)$ are given in case $(α\vee 1)ρ\neq 1$.