Confining gauge dynamics in a dark sector is promising to provide dark matter with a mass in the range of sub-GeV to GeV. Such dark sectors consist of composite particles such as dark baryons and dark mesons, that are neutral under the standard-model charge. A dark photon is introduced as a portal matter between the dark sector and the standard-model sector to alleviate cosmological problems (e.g., to maintain kinetic equilibrium between two sectors or to reduce the light dark-sector particles contributing to the dark radiation), and dark hadrons are produced through the same dark photon at accelerator-based experiments. As dark vector mesons and dark pions have similar masses, dark vector mesons can be long-lived particles, which will be explored by far-detector experiments. We study the future prospects of the LHC forward-detector experiments, FASER/FASER2 and FACET, for exploring the dark vector mesons. When the dark photon is heavier than the dark pions, the LHC forward-detector searches will be comparable to DarkQuest, and the invisible decay searches of dark photons will also explore the same parameter space. Meanwhile, when dark photons are lightest in the dark sector, their future prospects will be comparable to the visible decay searches for dark photons at LHCb, Belle-II, and HPS.