It's hard to tell with AncestryDNA because there is no chromosome browser. If possible, uploading everyone's raw data to Gedmatch and using their browser is the way to go.
This may or may not be a 3rd cousin. If you come from endogamous ethnic groups, a match like that is more likely to be a 4th or 5th cousin. But even if it is a 3rd cousin, that would mean you share great great grandparents. You have 16 of those. 8 lines total (I'm count the route to each couple as 1 line). 4 lines on your dad's side.
Do the relatives on your dad's side who tested cover all 4 lines? You can't say that the match isn't via a line just because there isn't a match with a relative from that line, because the match isn't close enough to be sure there would be matching segments. But it is more likely that your match with this person comes from one of your father's lines that isn't covered by family members who took DNA tests.