Sauntag,
PRONI has the following document: MIC/68
It’s a set of title deeds & testamentary papers relating to property at Glenwhirry. Graham, Taggart & Killen families. Dated 1725. Might be worth looking at.
Also I see in their names index that an Archibald Taggart of Glenwherry had a will probated in 1695; John in 1791 & Grisell Taggart alias Bell, of Glenwherry had one in probated in 1718. (She was likely a widow). All of those 3 will were destroyed in the 1922 fire. All that remains is the probate information given here.
You are up against what genealogists refer to the 1800 barrier in Irish research. It can often prove very difficult to get back much earlier than 1800 due to the absence of sufficient records. Did this Taggart family live in Glenwhirry townland, or Glenwhirry parish? If it was the parish do you know which townland. (I see 3 Taggart properties in Glenwhirry parish in Griffiths in 1862, in Glenhead, Jockeysquarter and Glenwhirry itself, so it’s important to identify the right family.)
What was James senior’s occupation, do you know?
Have you searched local graveyards for the family? If they were farmers, you may get a gravestone. If they were weavers or labourers you mostly won’t.