The three-nucleon force (3NF) plays a crucial role in many-nucleon systems. Specifically, it enhances the spin-orbit (SO) splitting, a key phenomenon in the manifestation of shell structure. However, its detailed mechanism has not been fully understood. Recently, we pointed out that a specific component, called the rank-1 component, originating from the two-pion exchange term of the 3NF, significantly contributes to the SO splitting. In this paper, we present shell-model results obtained by implementing the 3NF derived from chiral effective field theory, and discuss how the rank-1 component enhances the SO splitting in $p$-shell nuclei.