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Accreting black holes commonly exhibit hard X-ray emission, originating from a region of hot plasma near the central engine referred to as the corona. The origin and geometry of the corona are poorly understood, and models invoking either inflowing or outflowing material (or both) can successfully explain only parts of the observed phenomenology. In particular, recent works indicate that the time-averaged and variability property might originate in different regions of the corona. In this paper we present a model designed to move beyond the lamppost paradigm, with the goal of accounting for the vertical extent of the corona. In particular, we highlight the impact of including self-consistently a second lamppost, mimicking, for example, an extended jet base. We fully include the effect that the second source has on the time-dependent disk ionization, reflection spectrum, and reverberation lags. We also present an application of this new model to NICER observations of the X-ray binary MAXI J1820+070 near its hard-to-soft state transition. We demonstrate that in these observations, a vertically extended corona can capture both spectral and timing properties, while a single-lamppost model can not. In this scenario, the illumination responsible for the time-averaged spectrum originates close to the black hole, while the variability is likely associated with the ballistic jet.