Full-angle distributed Bragg reflectors (DBRs) consisting of numerous sub-DBRs with discrete central wavelengths have been developed to enhance performance of GaN-based flip-chip mini light-emitting diodes (FC mini-LEDs). However, relatively low reflectivity of full-angle DBRs at large angle incidence restricts further enhancement in performance of FC mini-LEDs. Here, we introduce a reflectivity optimization strategy for constructing high-reflectivity multiple-stack DBRs by rationally engineering the number of sub-DBRs and adjusting central wavelength distribution of sub-DBRs. Based on the reflectivity optimization strategy, we devise a Ti3O5/SiO2 quintuple-stack DBR which is composed of five sub-DBRs. Our quintuple-stack DBR maintains a high reflectivity (>97.5%) over a wide range of incident angles of light. Notably, compared with the full-angle DBR, our quintuple-stack DBR exhibits higher reflectivity at large angle incidence and thinner multilayer thickness. Furthermore, we demonstrate two types of GaN-based blue FC mini-LEDs with indium-tin oxide (ITO)/quintuple-stack DBR and ITO/full-angle DBR p-type ohmic contacts. Benefiting from superior reflection performance, blue FC mini-LED with ITO/quintuple-stack DBR achieves an enhancement of ~5.8% in light output power at 10 mA, in comparison with blue FC mini-LED with ITO/full-angle DBR. Our work signifies an advancement towards high-reflectivity DBRs, which enables higher-performance FC mini-LEDs.