We aim to study the spot evolution and differential rotation in the magnetically active cool K-type giant star sigma Gem from broadband photometry and continuous spectroscopic observations that span 150 nights. We use high-resolution, high signal-to-noise ratio spectra obtained with the Hertzsprung SONG telescope to reconstruct surface (photospheric) temperature maps with Doppler imaging techniques. The 303 observations span 150 nights and allow for a detailed analysis of the spot evolution and surface differential rotation. The Doppler imaging results are compared to simultaneous broadband photometry from the Tennessee State University T3 0.4m Automated Photometric Telescope. The activity from the stellar chromosphere, which is higher in the stellar atmosphere, is also studied using SONG observations of Balmer Halpha line profiles and correlated with the photospheric activity. The temperature maps obtained during eight consecutive stellar rotations show mainly high-latitude or polar spots, with the main spot concentrations above latitude 45{deg}. The spots concentrate around phase 0.25 near the beginning of our observations and around phase 0.75 towards the end. The photometric observations confirm a small jump in spot phases that occurred in February 2016. The cross-correlation of the temperature maps reveals rather strong solar-like differential rotation, giving a relative surface differential rotation coefficient of alpha=0.10+/-0.02. There is a weak correlation between the locations of starspots and enhanced emission in the chromosphere at some epochs.