The relationship between the triplet- and singlet-pair interaction coefficients in the uniform singlet- and triplet-pair extrapolation method recently suggested for extrapolating ab initio energies to the one-electron basis set limit is analyzed. Based on the premise that such a ratio is invariant over the configuration space of the molecule, generalizations of the method are investigated and their performance tested on extrapolations with MP2, CCD, CCSD, and MRCI(Q) energies. The best variant requires raw energies calculated using augmented correlated consistent basis sets with cardinal numbers up to X = 6 at a single geometry. A scheme is also suggested that performs better than the traditional X-3 law and possibly the original uniform singlet- and triplet-pair extrapolation method but requires data only up to X = Q.