Hi,
In my experience, these missing bits usually pop up when doing cross searches on family members. Naming coventions often give clues to family names for parents, brothers and sisters, aunts and uncles and connecting branches of family members.
Forward searches of connecting family members sometimes lead to a clue on origination.
Reverse searches starting with descendants of similar names in county Antrim can also lead to a more generalized area to search for clues.
Cemeteries are another option, as many families prefered a particular site if possible. Searches in the U.S. should look for offspring that died young while trying to cross search at LDS Family search as there were many diseaes events during the time period. Often these searches lead to a religious preference which helps to direct future searches. This helps to point the way to what Antrim records should be looked at.
As your ancestor was in the right time period for several wars in the U.S. where more data was collected than if seraching for someone in the 1700's, there should be records both local and military that might lead to the missing bits.
Good tips and links to some resources can be found at:
http://www.irish-genealogy-toolkit.com/Northern-Ireland-genealogy.html
Good Luck! Jeanette