We explore some topics in the phenomenology of gauge-mediated SUSY-breaking scenarios having a large hierarchy of Higgs VEVs, upsilon(U)/upsilon(D) = tan beta much greater than 1. Some motivation for this scenario is first presented. We then use a systematic, analytic expansion (including some threshold cor- rections) to calculate the mu-parameter needed for proper electroweak breaking and the radiative corrections to the B-parameter, which fortuitously cancel at leading order, If B = 0 at the messenger scale then tan beta is naturally large and calculable; we calculate it, We then confront this prediction with classical and quantum vacuum stability constraints arising from the Higgs-slepton potential, and indicate the preferred values of the top quark mass and messenger scale(s), The possibility of vacuum instability in a different direction yields an upper bound on the messenger mass scale complementary to the familiar bound from gravitino relic abundance. Next, we calculate the rate for b --> s gamma and show the possibility of large deviations (in the direction currently favored by experiment) from standard-model and small tan beta predictions. Finally, we discuss the implications of these findings and their applicability to future, broader and more detailed investigations. (C) 1997 Elsevier Science B.V.