Recent observations have reported that the Galactic globular clusters (GCs) with unusually extended horizontal-branch (EHB) morphologies show a significantly lower velocity dispersion compared with that of the entire Galactic GC system. We consider that the observed distinctive kinematics of GCs with EHB has valuable information on the formation epochs of GCs and accordingly discuss this observational result based on cosmological N-body simulations with a model of GC formation. We assume that GCs in galaxies were initially formed in low-mass haloes at high redshifts and we investigate final kinematics of GCs in their host haloes at z= 0. We find that GCs formed in haloes virialized at z > 10 show lower velocity dispersions on average than those formed at z > 6 for haloes with GCs at z= 0. We thus suggest that the origin of the observed lower velocity dispersion for the Galactic GCs with EHBs is closely associated with earlier formation epochs (z > 10) of haloes initially hosting the GCs in the course of the Galaxy formation. Considering that the origin of EHBs can be due to the presence of helium-enhanced second-generation stars in GCs, we discuss the long-standing second parameter problem of GCs in the context of different degrees of chemical pollution in GC-forming gas clouds within low-mass haloes virialized at different redshifts.