The Superconducting Kaon Spectrometer (SKS) magnet, which was in operation at the K6 beamline in the North Counter Hall of the KEK 12-GeV proton synchrotron from 1991 to 2005, was modified and moved to the newly constructed K1.8 beamline in the Hadron Hall at the Japan Proton Accelerator Research Complex (J-PARC). We reused the original coil assembly which includes coils and vessels, with a Gifford-MacMahon (GM) cryocooler for shield cooling. We used three Gifford-MacMahon/Joule-Thomson (GM/JT) cryocoolers instead of the previous 300-W helium refrigerator. The cooling power of a GM/JT cryocooler is 3.5 W at 4.5 K and 50 Hz. We also installed a new GM cryocooler to cool the GM/JT insert ports. Conventional copper current leads were replaced with Bi2223 high-temperature-superconductor (HTC) leads. We also required a new GM cryocooler to cool the HTC current leads. In total, we installed three GM/JT and three GM cryocoolers. We cooled the reconstructed SKS superconducting magnet at the K1.8 beamline from room temperature to 4.3 K over 1.5 months, applying LN2 and LHe directly. We confirmed that the three GM/JT cryocooles maintained the LHe level in the steady state. Helium gas pressure within the helium vessel was 113 kPa absolute and the condenser temperatures of the three GM/JT cryocoolers were 4.35 K, 4.46 K, and 4.35 K, respectively. We also measured the rise in temperature of the HTC leads when the magnet was continuously excited at a current of 400 A. The temperature of the HTC leads rose to 41 K and attained saturation after 5 hours. This was sufficiently safe for the Bi2223 HTC leads. Furthermore, we confirmed that two GM/JT cryocoolers could maintain the steady state; however, for safety reasons, we decided to use three GM/JT cryocoolers in the case of continuous excitation at more than 350 A.