The most sensitive method for detecting neutrinos at the very highest energies is the lunar Cherenkov technique, which employs the Moon as a target volume, using conventional radio telescopes to monitor it for nanosecond-scale pulses of Cherenkov radiation from particle cascades in its regolith. Multiple-antenna radio telescopes are difficult to effectively combine into a single detector for this purpose, while single antennas are more susceptible to false events from radio interference, which must be reliably excluded for a credible detection to be made. We describe our progress in excluding such interference in our observations with the single-antenna Parkes radio telescope, and our most recent experiment (taking place the week before the ICRC) using it in conjunction with the Australia Telescope Compact Array, exploiting the advantages of both types of telescope.