Can p-mode waves in sunspots propagate to the chromosphere and the corona? And what are their counterparts in different atmospheric heights? In order to study the connection between the photospheric p-mode waves and the waves observed above the photosphere, we use a helioseismic analysis technique, namely time-distance helioseismology, and analyze multi-height observations from different instruments. We find clear evidences that some p-mode waves in the photosphere, running penumbral waves in the chromosphere, and the periodic disturbances in the coronal fan structures are actually same magnetoacoustic waves that exhibit differently at the different atmospheric heights. The 6-mHz waves, with inclined wavefronts, propagate slantingly upward along magnetic field lines. The 3-mHz waves, forming backward-'C'-shape wavefronts, propagate mostly horizontally. Through numerical simulations, we demonstrate that these p-mode waves that can travel upward to the corona, possibly originate from sources located a few megameters beneath sunspots' surface.