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An efficient description of the structures of liquids and, in particular, the structural changes that happen with liquids on supercooling remains to be a challenge. The systems composed of soft particles are especially interesting in this context because they often demonstrate non-trivial local orders that do not allow to introduce the concept of the nearest-neighbor shell. For this reason, the use of some methods, developed for the structure analysis of atomic liquids, is questionable for the soft-particle systems. Here we report about our investigations of the structure of the simple harmonic-repulsive liquid in 3D using the triple correlation function (TCF), i.e., the method that does not rely on the nearest neighbor concept. The liquid is considered at reduced pressure ($P=1.8$) at which it exhibits remarkable stability against crystallization on cooling. It is demonstrated that the TCF allows addressing the development of the orientational correlations in the structures that do not allow drawing definite conclusions from the studies of the bond-orientational order parameters. Our results demonstrate that the orientational correlations, if measured by the heights of the peaks in the TCF, significantly increase on cooling. This rise in the orientational ordering is not captured properly by the Kirkwood's superposition approximation. Detailed considerations of the peaks' shapes in the TCF suggest the existence of a link between the orientational ordering and the slowdown of the system's dynamics. Our findings support the view that the development of the orientational correlations in liquids may play a significant role in the liquids' dynamics and that the considerations of the pair distribution function may not be sufficient to understand intuitively all the structural changes that happen with liquids on supercooling.