The lack of red and deep-red emitting molecular phosphors with high photoluminescence quantum yields remains a significant fundamental challenge and has implications in optoelectronic technologies for color displays and other consumer products. In this work, we introduce a series of seven new red or deep-red emitting heteroleptic bis-cyclometalated iridium(III) complexes, supported by five different ancillary ligands (L^X) from the salicylaldimine and 2-picolinamide families. Previous work had shown that electron-rich anionic chelating "L^X" ligands can be effective in supporting efficient red phosphorescence, and the complementary approach described here, in addition to being synthetically simpler, offers two key advantages over the previous designs. First, the "L" and "X" functionalities can be independently tuned, providing excellent control over the electronic energy levels and excited-state dynamics. Second, these classes of L^X ligands can have beneficial impacts on the excited-state dynamics but do not significantly perturb the emission color profile. Cyclic voltammetry experiments show that the substituents on the L^X ligand impact the HOMO energy but have a minimal effect on the LUMO energy. Photoluminescence measurements reveal that all the compounds luminesce in the red or deep-red region as a function of the cyclometalating ligand and exhibit exceptionally high photoluminescence quantum yields (ΦPL), comparable or superior to the best-performing red-emitting iridium complexes.