When a clip extent is set through the data frame properties, the Maplex/Standard labeling engine does not recognize the clip so particularlabels are are no longer visiblie or are clipped in half.
use the drawing toolbar tools to create a polygon graphic that corresponds to the area outside of the clipped extent. You can draw a large rectangle that is roughly the size of the unclipped dataframe, and a second polygon the corresponds to the clipped shape. Then use the Drawing >> Graphics Operations >> Remove overlap command. The resulting graphic will act as a labeling barrier and will prevent labels from being placed on the boundary and should eliminate the partial label problem.