  Consider a family of $\Delta$-latticed aperiodic random walks $\{S^{(a)},0\le
a\le a_0\}$ with increments $X_i^{(a)}$ and non-positive drift $-a$. Suppose
that $\sup_{a\le a_0}\mathbf{E}[(X^{(a)})^2]<\infty$ and $\sup_{a\le
a_0}\mathbf{E}[\max\{0,X^{(a)}\}^{2+\varepsilon}]<\infty$ for some
$\varepsilon>0$. Assume that $X^{(a)}\xrightarrow[]{w} X^{(0)}$ as $a\to 0$ and
denote by $M^{(a)}=\max_{k\ge 0} S_k^{(a)}$ the maximum of the random walk
$S^{(a)}$. In this paper we provide the asymptotics of
$\mathbf{P}(M^{(a)}=y\Delta)$ as $a\to 0$ in the case, when $y\to \infty$ and
$ay=O(1)$. This asymptotics follows from a representation of
$\mathbf{P}(M^{(a)}=y\Delta)$ via a geometric sum and a uniform renewal
theorem, which is also proved in this paper.
