Many problems in engineering and science involve calculation of difficult Laplace integrals of the form: I = integral(4)(0) t(alpha) exp(-xt(beta))F(t)dt In previous papers [Hanna and Davis (2011), Davis and Hanna (2013), referred to as Papers I and II, respectively], the authors introduced two new complementary analytical methods for the improved asymptotic (x -> infinity) approximation of these integrals when the upper limit A equal to infinity [the general Watson lemma (WL) problem for any A if x -> infinity]. A procedure is developed here that extends application of the previous improvement methods to the difficult problem of Laplace integrals having a finite upper limit (FUL) = A. In addition, problems having growing exponential behavior, certain infinite Fourier integrals and problems having large (t(alpha)) factors, are also considered. The main result is that, with some modifications, the exponential, expansion-point, and combination procedures developed for infinite integrals (Papers I and II) can be easily applied directly to FUL Laplace integrals. This is accomplished with the aid of a simple new "generalized incomplete gamma function (IGF)" algorithm which itself utilizes an improvement procedure. The new FUL procedure requires only F(t), F'(t), and a few terms of the Taylor expansion. A simple EXCEL program which implements the new procedure is discussed in detail in Appendix A and is freely available to users at http://www.d.umn.edu/similar to rdavis/CEC/. Many numerical comparisons presented here indicate that good engineering accuracy is achieved for these improved approximations at virtually all positive A values, over a very wide range in x, for various alpha, beta, and F(t) functions. Where comparisons are possible (A = infinity), the new results are far superior to those of the best Watson's lemma results.