We report a novel pressure(P)-indu.ced evolution on the magnetism in the P-induced superconductor CeRhIn5 revealed by measurements of the In-115 nuclear-spin-lattice-relaxation rate ((115)1/T-1). in the itinerant antiferromagnet CeRhIn5, with the Neel temperature T-N = 3.8 K at P=0, T-N begins to he reduced to T-N 3.6 K at P = 1,23 GPa that is close to the border of the superconducting phase, An emergent behavior is the observation of a pseudogap behavior that is revealed by a broad peak in the temperature dependence of 1/T1T at P = 1.23 GPa. With further increasing P up to P = 2.1 GPa where its bulk superconductivity is fully established, the pseudogap behavior is not seen in the normal state. The insensitive P dependence of T-c in CeRhIn5 is suggested to be related to that of the characteristic temperature T* which marks a localized to itinerant magnetic crossover.