The quadrupole resonator (QPR) is a dedicated sample-test cavity for the RF characterization of superconducting samples in a wide temperature, RF field, and frequency range. Its main purpose is high resolution measurements of the surface resistance with direct access to the residual resistance, thanks to the low frequency of the first operating quadrupole mode. In addition to the well-known high resolution of the QPR, a bias of measurement data toward higher values has been observed, especially in higher harmonic quadrupole modes. Numerical studies show that this can be explained by parasitic RF losses on the adapter flange used to mount samples into the QPR. Coating several micrometers of niobium on those surfaces of the stainless steel flange that are exposed to the RF fields significantly reduced this bias, enabling a direct measurement of a residual resistance smaller than 5 nΩ at 2 K and 413 MHz. A constant correction based on simulations was not feasible due to deviations from one measurement to another. However, this issue is resolved given these new results.