Rigid water column (RWC) models simulate the unsteady-incompressible hydraulics of pressurized pipe networks. They conceptually lie between water hammer and quasi-steady models, yet despite their intrinsic strengths, existing RWC formulations suffer efficiency-, stability-, and interpretation-related challenges; thus, they are typically overlooked as a modeling alternative. To address the aforementioned limitations, this article presents the RWC global gradient algorithm (GGA), a novel formulation for pipe networks that has greater efficiency and overcomes the numerical challenges. The RWC GGA extends the generalized GGA (G-GGA) to consider inertial effects in addition to variable-area tanks and mixed (i.e., demand and pressure-dependent) outflows. Two pipe networks of simple and moderate complexity are used to compare the new approach against two other RWC algorithms, the G-GGA, and a water hammer model: the current work is shown to have improved stability and efficiency relative to previous work. The RWC GGA is also found to have a computational cost only slightly greater than that of the G-GGA for the same time-step size. Overall, this work highlights the practical utility of RWC models to simulate slow transient events and controlled operations.