In a previous article, the authors introduced a new method for the improved asymptotic evaluation of Laplace integrals of the type integral(infinity)(0) t(alpha) exp(-xt)F(t(k))dt for large x, where k must be 1 or 2 (Hanna and Davis, 2011). This method (called optimal exponential or OE) also applies to the acceleration of regular perturbation and power series. Here we develop a new analytical procedure (called expansion point or XP) to evaluate more general problems of the type integral(A)(0) p(t)F(t)dt. This includes the most general Watson lemma problem of the type integral(A)(0) t(alpha) exp(-xt(beta))F(t)dt, which cannot be handled by the OE method. The XP method is simple and general and offers good practical accuracy. It is especially powerful when combined with an "extended" exponential procedure. Application of the new method is illustrated with several problems important in chemical engineering.