Stellar feedback, expanding H ii regions, wind-blown bubbles, and supernovae are thought to be important triggering mechanisms of star formation. Stellar associations, being hosts of significant numbers of early-type stars, are the loci where these mechanisms act. In this part of our photometric study of the star-forming region NGC 346/N66 in the Small Magellanic Cloud, we present evidence based on previous and recent detailed studies that it hosts at least two different events of triggered star formation, and we reveal the complexity of its recent star formation history. In our earlier studies of this region (Papers I and III) we find that besides the central part of N66, where the bright OB stellar content of the association NGC 346 is concentrated, an arclike nebular feature, north of the association, hosts recent star formation. This feature is characterized by a high concentration of emission-line stars and young stellar objects, as well as embedded sources seen as IR emission peaks that coincide with young compact clusters of low-mass pre-mainsequence stars. All these objects indicate that the northern arc of N66 encompasses the most current star formation event in the region. We present evidence that this star formation is the product of a different mechanism than that in the general area of the association, and that it is triggered by a wind-driven expanding H ii region (or bubble) blown by a massive supernova progenitor, and possibly other bright stars, a few megayears ago. We propose a scenario according to which this mechanism triggered star formation away from the bar of N66, while in the bar of N66 star formation is introduced by the photoionizing OB stars of the association itself.