We propose and demonstrate evanescently-decoupled, solid-angle-optimized distributed Bragg reflectors (DBRs) for AlGaInP light-emitting diodes (LEDs). The thickness of each DBR layer is tuned to the wavelength slightly longer than the emission peak of the active medium in order to maximize the radiated power integrated over the top surface. In addition, to increase the horizontal radiation through the side facets, the glancing-angle reflectivity at the AlInP/AlAs interface is improved by employing an AlAs layer thicker than the attenuation length of the evanescent field. With the improved DBR, the integrated output power of AlGaInP LEDs is enhanced by a factor of 1.9 in comparison to those of LEDs with conventional DBRs. Additional 1.25-fold enhancement is observed by incorporating an square-lattice hole array (a=1200nm) into the top GaP surface by a conventional photolithography.