With an increasing interest in hydrogen as an alternative fuel for transportation, there is a need to develop tools for the prediction of ignition events. A cost-effective passive scalar formulation has been recently developed to predict hydrogen auto-ignition. A single scalar advection-diffusion-reaction equation is used to reproduce the chain-branched ignition process, where the scalar represents the radical pool responsible of ignition (H, O, OH, HO2 , H2O2). The scalar reaction rate is analytically deduced from the Jacobian matrix associated to hydrogen ignition chemistry. This method was found to reproduce with good accuracy the ignition delays obtained by detailed chemistry for temperature where the branching is the leading process. For temperature close or below the crossover temperature, where other phenomenon such as the thermal runaway are important, the scalar approach fails to predict correctly ignition events. Thus, an extension of the scalar source term formulation is proposed to extend its validity over the entire temperature range. In addition, a simple way to approximate the diffusion properties of the scalar is introduced: the radical pool composition may vary drastically, with molecules having very different diffusion properties (e.g. H and HO2). The complete modified framework is presented and its capability is assessed in canonical scenarios and more complex simulations relevant to hydrogen safety.