I'm glad that you've found more on your ancestors. Maybe you could post an update to your previous query, to help others who may browse this site looking for the same families.
You probably know that you can order an instantly-viewable transcript copy of Margaret Kane's 1869 Strabane District death record (and many other records) from the Irish Family History Foundation at http://tyronefermanagh.rootsireland.iehttp://tyronefermanagh.rootsireland.ie. And you can get a copy of an original handwritten birth/marriage/death certificate from the Dublin GRO (the 1869 death index entry you cited is from the LDS FamilySearch database of Irish Civil Registrations).
Ages are sometimes incorrect in censuses, death records and other documents, but, even taking that into account, I think that Margaret Kane who died in 1869 at age 41 is still a bit too young to be the mother of Martha Kane born 1820s-1830s. Margaret's age might be "off" a few years when stated for her death certificate, but not several decades. She might be a sister of Martha Kane (born in right time period), or the wife of a Kane man who was Martha's brother. You could check the valuation revisions to see who took over Margaret's farm after she was counted in Griffith's Valuation. She was actually counted at Dullerton in Donagheady parish, the same home location as the Hasletts, which does suggest a connection. The revision books are at PRONI.
I think I've already mentioned that you should check all records on a surname in the district or parish of interest, so I'd check vital record indexes for ALL Kanes and Hazletts in Strabane District.
Back to CHURCH RECORDS: Donaghedy's 1st Presbyterian Church registers have different dates from 2nd Presbyterian Church registers. The original Donagheady Presbyterian church congregation had a dispute and split into two separate congregations in 1741. The group that continued to meet in the old church was called "1st Donagheady," while the new congregation came to be known as "2nd Donagheady." The congregations were reunited in 1933. Very early records of those churches have not survived. The data from 2nd Donagheady Presbyterian Church Marriages, 1838-1845, as listed on the Co. Tyrone web site, was transcribed by the generous genealogist, Faye Logue, from PRONI microfilm MIC 1/P/459.
As you see in PRONI's church records guide to DONAGHEDY parish, CO. TYRONE:
Church of Ireland, Donagheady (Derry diocese) -- MIC1/35; CR/1/26
Baptisms, 1697-1723, 1753-65, 1818-19 and 1826-74;
marriages, 1697-1726, 1754-64 and 1826-44; marriage
licences, 1817 and 1829-53; burials, 1698-1726, 1754-
7 and 1826-89; vestry minutes, 1697-1723 and 1754-
1919; accounts, 1829-1922; confirmation lists, 1872,
1875, 1877, 1880, 1883 and 1886.
In local custody: Baptisms, 1874-; marriages, 1845-; burials, 1889-; churchwardens’ account books, 1793 and 1871-; preachers’ books, 1855-.
Presbyterian, 1st Donagheady -- MIC1P/49/2; MIC1P/458
Baptisms, 1875-1932; session minutes, 1876-1932; committee minutes, 1869-71 and 1883-1911; congregational account books, 1867-1932; financial statements, 1870 and 1872-3; maps of Grange graveyard, c.1921; collecting books for poor rate, 1856 and 1861.
In local custody: Marriages, 1860-; stipend books, 1804-60, 1867-96 and 1898-1914.
Presbyterian, 2nd Donagheady -- MIC1P/49/2; MIC1P/459
Baptisms, 1838-1976; marriages, 1838-44; session minutes, 1832-1933; committee minutes, 1882-1933; Financial statements, 1866, 1870, 1872 and 1874; stipend books, 1832-56 and 1874-1915; communicants’ rolls, 1865-1937.
In local custody: Marriages, 1845-.
Presbyterian, Donemana -- MIC1P/425
Baptisms, 1856-1943; marriages, 1845-1926.
There's also a Reformed Presbyterian church at Bready whose records have been microfilmed:
Reformed Presbyterian, Bready -- MIC1C/11
Baptisms, 1866-1984; marriages, 1864-1936; session minutes, 1872-1932.
Several extensive indexes of Donagheady church registers are on the Bready Ancestry site: www.breadyancestry.com. I did see additional Hazlett/Haslett/Heaslett entries there but didn't note them for you. Look at all the Bready databases; very interesting history and helpful genealogical indexes.
A small but comprehensive Irish church records guide is Brian Mitchell's source book: A Guide to Irish Parish Registers (Genealogical Publishing Company).
Also the Presbyterian Historical Society in Belfast has Presbyterian congregation records that may not be found elsewhere.
In my previous message post, I mentioned the TITHE books not surviving for Donagheady parish. However the Valuation Field Books are available and date from 1835, with householders' names, field notes, valuation of houses, etc. Those books are at PRONI under land records. Unfortunately, there's no on-line index to them (yet) that I know of. You'd have to have someone research them at PRONI.