Permanent electric dipole moments (EDMs) of fundamental particles such as the electron are signatures of parity and time-reversal violation occurring in physics beyond the standard model. EDM measurements probe new physics at energy scales well beyond the reach of present-day colliders. Recent advances in assembling molecules from ultracold atoms have opened up new opportunities for improving the reach of EDM experiments. However, the magnetic field sensitivity of such ultracold molecules means that new measurement techniques are needed before these opportunities can be fully exploited. We present a technique that takes advantage of magnetically insensitive hyperfine clock transitions in polar molecules, offering a way to improve both the precision and accuracy of EDM searches with ultracold assembled molecules.