We present a newly-developed microwave probe for performing sensitive high-field/multi-frequency electron spin resonance (ESR) measurements under high hydrostatic pressures. The system consists of a BeCu-made pressure-resistant vessel, which accommodates the investigated sample and a diamond microwave coupling window. The probe's interior is completely filled with a pressure-transmitting fluid. The setup operates in reflection mode and can easily be assembled with a standard oversized microwave circuitry. The probe-head withstands hydrostatic pressures up to 1.6 GPa and interfaces with our home-built quasi-optical high-field ESR facility, operating in a millimeter/submillimeter frequency range of 105-420 GHz and in magnetic fields up to 16 T. The overall performance of the probe was tested, while studying the pressure-induced changes in the spin-relaxation mechanisms of a quasi-1D conducting polymer, KC(60). The preliminary measurements revealed that the probe yields similar signal-to-noise ratio to that of commercially available low-frequency ESR spectrometers. Moreover, by observing the conduction electron spin resonance (CESR) linewidth broadening for KC(60) in an unprecedented microwave frequency range of 210-420 GHz and in the pressure range of up to 1.6 GPa, we demonstrate that a combination of high-pressure ESR probe and high-field/multi-frequency spectrometer allows us to measure the spin relaxation rates in conducting spin systems, like the quasi-1D conductor, KC(60).