Based on the ground-breaking work of J.Vignaux [Vi] from 1939, a basic theory of asymptotic Laplace transforms applicable to locally Bochner integrable, Banach space valued functions on finite intervals [0, T) or on the half line [0, ∞) with arbitrary growth at infinity was recently developed by the present authors in [L-N]. However, several matters of considerable importance remained unclear, unsettled, or unexplored. In this paper we will focus on the following topics.