Lead-halide perovskite nanocrystals (NCs) have just emerged as a novel type of semiconductor nanostructure possessing great potentials in the optoelectronic, photovoltaic and quantum-information-processing applications. This renders it extremely necessary to have a comprehensive understanding of their electronic energy-level structures, which mysteriously exhibit either a doublet or a triplet exciton peak at the single-particle level. Here we show that transition from doublet to triplet excitons in single CsPbI3 NCs can be triggered by reinforcing quantum confinement in the same batch of sample upon being stored in the ambient environment. Besides size reduction and blue-shifted emission, this enhanced quantum confinement is also manifested by the suppressed emission of multiple and charged excitons in single CsPbI3 NCs with a triplet-exciton configuration. We propose that the doublet and triplet excitons should correspond respectively to the weak and strong quantum confinement regimes of single CsPbI3 NCs, with the electron-hole exchange interaction and the Rashba effect determining the exact energy-level alignments and the fine-structure splitting values.