Because oligonucleotides are short sequences of nucleic acid bases, their association in solution with complementary strands (hybridization) is often seen to conform to a simple two-state model. However, experimental evidence suggests that, despite their short length, oligonucleotides may hybridize through multiple states involving intermediates. We investigate whether these apparently contradictory scenarios are possible by imposing different levels of sequence specificity on a lattice model of oligonucleotides in solution, which we introduced in Part I [J. C. Araque et al., J. Chem. Phys. 134, 165103 (2011)]. We find that both multiple-intermediate (weakly cooperative) and two-state (strongly cooperative) transitions are possible and that these are directly linked to the level of sequence specificity. Sequences with low specificity hybridize (base-by-base) by way of multiple stable intermediates with increasing number of paired bases. Such intermediate states are weakly cooperative because the energetic gain from adding an additional base pair is outweighed by the conformational entropy loss. Instead, sequences with high specificity hybridize through multiple metastable intermediates which easily bridge the configurational and energetic gaps between single- and double-stranded states. These metastable intermediates interconvert with minimal loss of conformational entropy leading to a strongly cooperative hybridization. The possibility of both scenarios, multiple- and two-states, is therefore encoded in the specificity of the sequence which in turn defines the level of cooperativity.