An interrupted time series with a parallel control group (ITS-CG) design is a powerful quasi-experimental design commonly used to evaluate the effectiveness of an intervention, on accelerating uptake of useful public health products, and can be used in the presence of regularly collected data. This paper illustrates how a segmented Poisson model that utilizes general estimating equations (GEE) can be used for the ITS-CG study design to evaluate the effectiveness of a complex social accountability intervention on the level and rate of uptake of modern contraception. The intervention was gradually rolled-out over time to targeted intervention communities in Ghana and Tanzania, with control communities receiving standard of care, as per national guidelines. Two ITS GEE segmented regression models are proposed for evaluating of the uptake. The first, a two-segmented model, fits the data collected during pre-intervention and post-intervention excluding that collected during intervention roll-out. The second, a three-segmented model, fits all data including that collected during the roll-out. A much simpler difference-in-difference (DID) GEE Poisson regression model is also illustrated. Mathematical formulation of both ITS-segmented Poisson models and that of the DID Poisson model, interpretation and significance of resulting regression parameters, and accounting for different sources of variation and lags in intervention effect are respectively discussed. Strengths and limitations of these models are highlighted. Segmented ITS modelling remains valuable for studying the effect of intervention interruptions whether gradual changes, over time, in the level or trend in uptake of public health practices are attributed by the introduced intervention.