The high-mass young stellar object G358.93-0.03 underwent an accretion burst during the period from 2019 January to June. Given its extraordinary conditions, a number of new maser transitions may have been naturally excited during the burst stage. Searching for new maser lines and monitoring maser variability associated with the accretion burst event are important for understanding the complex conditions of the massive star formation toward G358.93-0.03. In this work, using the Shanghai 65m Tianma Radio Telescope, we continuously monitored the multiple maser (including methanol and water) transitions toward G358.93-0.03 during the burst in the period from 2019 March 14 to May 20. There were 23 CH3OH maser transitions and one H2O maser transition detected from the monitoring. Nearly all the detected maser transitions toward this source have dramatic variations in their intensities within a short period of ~2 months. Eight new methanol transitions from G358.93-0.03 were identified to be masering in our observations based on their spectral profile, line width, intensity, and the rotation diagram. During the monitoring, the gas temperature of the clouds in the case of saturated masers can show a significant decline, indicating that the maser clouds were going through a cooling process, possibly associated with the propagation of a heat wave induced by the accretion burst. Some of the maser transitions were even detected with the second flares in 2019 April, which may be associated with the process of the heat-wave propagation induced by the same accretion burst acting on different maser positions.