In this paper, the properties of the omnidirectional photonic bandgap (OBG) realized by one-dimensional (1D) photonic crystals with a staggered structure which is composed of superconductor and isotropic dielectric have been theoretically investigated by the transfer matrix method (TMM). From the numerical results, it has been shown that such OBG is insensitive to the incident angle and the polarization of electromagnetic wave (EM wave), and the frequency range and central frequency of OBG can be tuned by the ambient temperature of system, the average thickness of superconductor layer, the average thickness of dielectric layer, and staggered parameters, respectively. The bandwidth of OBG can be notably enlarged with increasing average thickness and staggered parameter of superconductor layer. Moreover, the frequency range of OBG can be narrowed with increasing the average thickness, staggered parameter of dielectric layer, and ambient temperature, respectively. The damping coefficient of superconductor layer has no effect on the bandwidth of OBG under low-temperature conditions. It is shown that 1D superconductor–dielectric photonic crystals (SDPCs) have a superior feature in the enhancement of frequency range of OBG. This kind of OBG has potential applications in filters, microcavities, and fibers, etc.