We utilize dynamic light scattering (DLS) and passive microrheology to examine the phase behavior of a supramolecular polymer at very high pressures. The monomer, 2,4-bis(2-ethylhexylureido)toluene (EHUT), self-assembles into supramolecular polymeric structures in the nonpolar solvent cyclohexane by means of hydrogen bonding. By varying the concentration and temperature at atmospheric pressure, the formation of the viscoelastic network (at lower temperatures) and predominantly viscous phases, based on self-assembled tube and filament structures, respectively, has been established. The associated changes in the rheological properties have been attributed to a structural thickness transition. Here, we investigate the effects of pressure variation from atmospheric up to 1 kbar at a given concentration. We construct a temperature-pressure diagram that reveals the predominance of the viscoelastic network phase at high pressures. The transition from the viscoelastic network organization of the tubes to a weaker viscous-dominated structure of the filaments is rationalized by using the Clapeyron equation, which yields an associated volume change of about 8 Å3 per EHUT molecule. This change is further explained by means of Molecular Dynamics simulations of the two phases, which show a decrease in the molecular volume at the filament-tube transition, originating from increased intermolecular contacts in the tube with respect to the filament. These findings offer insights into the role of pressure in stabilizing self-assemblies.