ADAPT-VQE stands out as a robust algorithm for constructing compact ansätze for molecular simulation. It enables to significantly reduce the circuit depth with respect to other methods, such as UCCSD, while achieving higher accuracy and not suffering from so-called barren plateaus that hinder the variational optimisation of many hardware-efficient ansätze. In its standard implementation, however, it introduces a considerable measurement overhead in the form of gradient evaluations trough estimations of many commutator operators. In this work, we mitigate this measurement overhead by exploiting a recently introduced method for energy evaluation relying on Adaptive Informationally complete generalised Measurements (AIM). Besides offering an efficient way to measure the energy itself, Informationally Complete (IC) measurement data can be reused to estimate all the commutators of the operators in the operator pool of ADAPT-VQE, using only classically efficient post-processing. We present the AIM-ADAPT-VQE scheme in detail, and investigate its performance with several H4 Hamiltonians and operator pools. Our numerical simulations indicate that the measurement data obtained to evaluate the energy can be reused to implement ADAPT-VQE with no additional measurement overhead for the systems considered here. In addition, we show that, if the energy is measured within chemical precision, the CNOT count in the resulting circuits is close to the ideal one. With scarce measurement data, AIM-ADAPT-VQE still converges to the ground state with high probability, albeit with an increased circuit depth in some cases.