We report on a detailed diagnostic study of the extremely efficient supersonic chemical oxygen-iodine laser (COIL) recently developed in our laboratory (Appl. Phys. Lett., 85, 5851 (2004)). 40.0% efficiency was measured for 1 s at the early stage of operation, followed by a sustained 35.5% chemical efficiency for 20 s. The power and spatial distributions of the gain and temperature across the flow were measured for different supersonic nozzles with both staggered and non-staggered iodine injection holes, different injection locations along the flow and nozzle throat-heights. 0.73 yield of singlet oxygen was obtained for conditions corresponding to the highest efficiency. The effects of the partial pressure of O2 and the residence time of the flow in the generator, as well as the heating of the nozzle, are shown to be crucial in attaining this high efficiency.