What is the process of concrete harding called after the initial setting process is over?

The process that occurs after initial set is commonly referred to as Curing! What actually happens is "Hydration" , which is the reaction that takes place within the concrete matrix between the chemicals and the water in the mix.

Very important that mix water be retained inside for complete hydration to take place. Of late, engineers an design professionals are placing much more emphasis on "Internal Curing" than ever before!