In this paper we continue our study of exit times for random walks with independent but not necessarily identical distributed increments. Our paper "First-passage times for random walks with non-identically distributed increments" was devoted to the case when the random walk is constructed by a fixed sequence of independent random variables which satisfies the classical Lindeberg condition. Now we consider a more general situation when we have a triangular array of independent random variables. Our main assumption is that the entries of every row are uniformly bounded by a constant, which tends to zero as the number of the row increases.