I'm not sure about what're you precisely asking. Some intermetallic phases are capable of order themselves during a slow cooling. This transformation is not complete, and depends on the temperature. The more intense you cool (the stronger the undercooling is), the lower the percentage you get.
Yes, the cooling rate affects the order to disorder transition since different rates impose different relaxation times of kinetics phase transformations and so controlling the formation and evolution of the two competing phases. Therefore, solidification during high cooling rate leads to the preferred selection of disordered phase and need long time to relaxes before the subsequent evolution leads to the aging and to continuous transition to a solid ordered phase out of any critical under-cooling concept.