Lead-free halide double perovskites (HDPs) have recently been proposed as potential stable and environment-friendly alternatives to lead-based halide perovskites. Bandgap engineering plays a vital role in the optoelectronic applications of HDP materials. In this study, methods combining high-pressure techniques with density functional theory calculations were employed to implement the bandgap engineering of a classic HDP-based (NH₄)₂SnBr₆. Under high pressure, (NH₄)₂SnBr₆ exhibits a redshift of the bandgap with increasing pressure up to 6.3 GPa and a sudden blueshift up to 20.2 GPa, followed by a redshift at higher pressures, which is relevant to the cubic–tetragonal phase transition, direct–indirect transition, and amorphization, respectively. Our results enrich the understanding of the structural–optical properties of (NH₄)₂SnBr₆ and reveal the special role of NH₄⁺ cations in pressure-induced bandgap engineering, thus providing important information for application in optoelectronic devices and helping to design ideal materials with higher efficiency.