The altermagnet candidate MnTe has recently gained significant interest due to its unconventional magnetic ordering. One of the key features of altermagnetism is the momentum-dependent spin-split band and its temperature-dependent evolution. Yet a fully momentum-resolved experimental investigation, including out-of-plane direction, is still lacking. Here, we systematically investigate the electronic structure of epitaxially grown MnTe by using angle-resolved photoemission spectroscopy (ARPES). Our photon-energy-dependent ARPES data reveal significant out-of-plane dispersions consistent with previous theoretical calculations. More interestingly, we identify two distinct temperature-dependent electronic band structure evolutions at different out-of-plane momentum positions: momentum-dependent energy shifts at the nodal plane and substantial spectral weight suppression at the off-nodal plane. These findings may suggest the importance of considering both the itinerant and localized nature of the magnetic ordering and momentum-dependent interactions. Our work provides crucial insights into the complex correlation between momentum, temperature, and electronic structure in MnTe, contributing to a deeper understanding of altermagnetism.