The present work investigates the joined effects of temperature and static magnetic field on the electrical properties of a 64 MHz planar high-temperature superconducting (HTS) coil, in order to enhance the signal-to-noise ratio (SNR) in nuclear magnetic resonance (NMR) applications with a moderate decrease of the HTS coil temperature (T(HTS)). Temperature control is provided with accuracy better than 0.1 K from 80 to 66 K by regulating the pressure of the liquid nitrogen bath of a dedicated cryostat. The actual temperature of the HTS coil is obtained using a straightforward wireless method that eliminates the risks of coupling electromagnetic interference to the HTS coil and of disturbing the static magnetic field by DC currents near the region of interest. The resonance frequency (f0) and the quality factor (Q) of the HTS coil are measured as a function of temperature in the 0-4.7 T field range with parallel and orthogonal orientations relative to the coil plane. The intrinsic HTS coil sensitivity and the detuning effect are then analyzed from the Q and f0 data. In the presence of the static magnetic field, the initial value of f0 in Earth's field could be entirely recovered by decreasing T(HTS), except for the orthogonal orientation above 1 T. The improvement of Q by lowering T(HTS) was substantial. From 80 to 66 K, Q was multiplied by a factor of 6 at 1.5 T in orthogonal orientation. In parallel orientation, the maximum measured improvement of Q from 80 K to 66 K was a factor of 2. From 80 to 66 K, the improvement of the RF sensitivity relative to the initial value at the Earth's field and ambient pressure was up to 4.4 dB in parallel orientation. It was even more important in orthogonal orientation and continued to increase, up to 8.4 dB, at the maximum explored field of 1.5 T. Assuming that the noise contributions from the RF receiver are negligible, the SNR improvement using enhanced HTS coil cooling in NMR experiments was extracted from Q measurements either with or without the presence of the sample. Notably, the additional cooling in the presence of conductive samples appears more beneficial at higher field strengths and with an orthogonal incidence than with parallel. The temperature range accessible here, involving a relatively straightforward cryogenic design, brings a gain in RF sensitivity that is of great significance to cutting-edge applications with very weakly conducting samples, small biological specimens, or small animals in vivo. This work also demonstrates a better tolerance to thin-film orientation misalignments relative to the magnetic field, and this could eventually play a role in designing effective non-planar HTS coils or coil arrays which include elements of various orientations. Finally, the data provided in this work may help understand some critical aspects in the design of HTS coils for NMR and MRI applications and accounts for the presence of the static magnetic field, particularly regarding the SNR loss due to a decreased quality factor and detuning issues.