Detonation initiation is a prerequisite to normal operations of an oblique detonation engine (ODE), and initiation-assistant measures are imperative in cases of initiation failure that occur in a length-limited combustor under wide-range flight conditions. This study numerically investigates the initiation characteristics of oblique detonation waves (ODWs) in H₂-fueled ODE combustors at wide-range flight Mach numbers Ma_f or flight altitudes H_f. Failures of ODW initiation are observed at both low Ma_f and high H_f if no measure is taken to assist initiation. Through analyses of the flow fields and theoretical predictions of the ignition induction length L_ind, the data reveal that the detonation failure at low Ma_f is raised by the significant decrease in the post-shock temperature due to insufficient shock compression, leading to a significant increase in L_ind. The detonation failure at high H_f is caused by the rapid decrease in the combustor inflow pressure as H_f increases, which also results in an increase in L_ind. With further identifications of the key flow structures crucial to detonation initiation, an initiation-assistant concept employing a transverse H₂ jet is proposed. The simulation results show that through an interaction between the incident oblique shock wave and the jet shock wave, the transverse jet helps to initiate an ODW in the combustor at a low Ma_f, and the initiation location is relatively fixed and determined by the jet location. At high H_f, a Mach reflection pattern is formed in the combustor under the effects of the transverse jet, and detonative combustion is achieved by the generated Mach stem and its reflected shock waves. The proposed concept of using transverse jets to assist detonation initiation provides a practical reference for future development of ODEs that are expected to operate under wide-range flight conditions.