Accurate numerical solutions of the equations of hydrodynamics play an ever more important role in many fields of astrophysics. In this work, we reinvestigate the accuracy of the moving-mesh code textsc{Arepo} and show how its convergence order can be improved for general problems. In particular, we clarify that for certain problems textsc{Arepo} only reaches first-order convergence for its original formulation. This can be rectified by simple modifications we propose to the time integration scheme and the spatial gradient estimates of the code, both improving the accuracy of the code. We demonstrate that the new implementation is indeed second-order accurate under the $L^1$ norm, and in particular substantially improves conservation of angular momentum. Interestingly, whereas these improvements can significantly change the results of smooth test problems, we also find that cosmological simulations of galaxy formation are unaffected, demonstrating that the numerical errors eliminated by the new formulation do not impact these simulations. In contrast, simulations of binary stars followed over a large number of orbital times are strongly affected, as here it is particularly crucial to avoid a long-term build up of errors in angular momentum conservation.
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