Zero-field and longitudinal-field muon-spin-relaxation measurements have been performed in partially Fe-substituted La2−xSrxCu1−yFeyO4 in a wide range of hole concentration, to investigate the magnetic state induced by the Fe substitution recently suggested from the neutron-scattering measurements [Phys. Rev. Lett. 107, 127002 (2011)]. It has been found that the magnetic transition temperature is notably enhanced through the 1% Fe substitution in a wide range of hole concentration where superconductivity appears in Fe-free La2−xSrxCuO4. In the underdoped regime, the Fe-induced magnetic order can be understood in terms of the concept of stripe pinning by Fe as in the case of the Zn-induced one in La2−xSrxCu1−yZnyO4. In the overdoped regime, on the other hand, the Fe-induced magnetic order is short-ranged, which is distinct from the stripes. It is plausible that a spin-glass state of Fe spins derived from the Ruderman-Kittel-Kasuya-Yosida interaction is realized in the overdoped regime, suggesting a change of the ground state from the strongly correlated state to the Fermi-liquid state with hole doping in La-214 high-Tc cuprates.