We simultaneously incorporate two common extensions of the hadron resonance gas model, namely the addition of extra, unconfirmed resonances to the particle list, and the excluded volume repulsive interactions. We emphasize the complementary nature of these two extensions and identify combinations of conserved charge susceptibilities that allow us to constrain them separately. In particular, ratios of second-order susceptibilities like χ11BQ/χ2B and χ11BS/χ2B are sensitive only to the baryon spectrum, while fourth-to-second order ratios like χ4B/χ2B, χ31BS/χ11BS, or χ31BQ/χ11BQ are mainly determined by repulsive interactions. Analysis of the available lattice results suggests the presence of both the extra states in the baryon-strangeness sector and the repulsive baryonic interaction, with indications that hyperons have a smaller repulsive core than nonstrange baryons. The modified hadron resonance gas model presented here significantly improves the description of lattice QCD susceptibilities at chemical freeze-out and can be used for the analysis of event-by-event fluctuations in heavy-ion collisions.