By extending the framework of the gravitational clock compass we show how a suitably prepared set of clocks can be used to extract information about the gravitational field in the context of general relativity. Conceptual differences between the extended and the standard clock compass are highlighted. Particular attention is paid to the influence of kinematic quantities on the measurement process and the setup of the compass. Additionally, we present results of simulations of the inference process for the acceleration and the curvature components. Several examples of different strategies for the computation of the posterior probability distributions of the curvature components are discussed. This allows us to anticipate the precision with which physical quantities could be determined in a realistic measurement.