The quartet states of the SO+ molecular ion are described theoretically using the internally contracted multireference configuration interaction approach and natural orbitals generated from a state-averaged density matrix. Correlation-consistent polarized-valence quadruple-zeta atomic functions are used in the expansion of the one-electron basis. Potential energy curves are presented for all the states, and solutions of the radial Schrodinger equation allowed the determination of vibrational energy differences and spectroscopic constants. For the b (4)Sigma- state, this study corroborates the available experimental data and extends the spectroscopic information to regions not yet accessed experimentally; an alternative explanation for the predissociation mechanism is also suggested, For the a (II)-I-4 state, our data and analysis are indicative that the vibrational spectroscopic constants derived from the photoelectron spectra might be underestimated. It also leaves open the possibility that the experimental vibrational level numbering might have to be increased by one unit. Transition probabilities as given by the Einstein A coefficients, and Franck-Condon factors are also provided to help analyze the experimental data. Of immediate relevance to the direct ion-fragment spectroscopy, this study predicts the existence of a new bound (II)-I-4 state in the energy range of photons used in these experiments. This new state crosses the b (4)Sigma(-) curve very close to where it was supposed to be crossed by the 1 (4)Sigma(+), and its repulsive side runs almost parallel to this latter state. Our theoretical prediction places the 1 (4)Sigma(+) State still lower than it was inferred experimentally. For the a (II)-I-4-1 (4)Sigma(+) transition we have also computed the transition moment function and showed that its constancy assumed in the simulation of the experimental intensity data is not valid.