The Remo, Rabbi Moshe Isserles, writes that there are indeed places where the accepted custom is for the Shaliach Tzibbur who leads the Slichot to continue and lead the rest of the day’s services. The Mishna Berura explains that this is based on the concept that “He who begins a mitzva is told – finish”. This holds true for the evening service that is ahead of the Slichot that are said in the evening, but not the evening service after the following sundown.