Heavy fermion compounds consisting of two or more inequivalent local moment sites per unit cell have been a promising platform of investigating the interplay between distinct Kondo screenings that is absent in the conventional systems containing only one rare-earth ion per unit cell. We report a remarkable enhancement of the antiferromagnetic (AF) ordering tendency in the staggered periodic Anderson model (PAM) with two alternating inequivalent local moments if their hybridization strengths reside in the Kondo singlet and antiferromagnetic insulator regime separately of the phase diagram of homogeneous PAM. Our results uncover the rich physics induced by the interplay of multiple energy scales in the staggered PAM and furthermore implies the ubiquitous existence of the enhancement of physical quantities in general inhomogeneous systems.