Hello Marie,
Re Edmund's birth in Tyrone - is it possible that his maternal grandparents lived there? I note from some records I've collected that a Rev. John Jephson lived in Mullabrack, Co. Armagh in the mid 1820s. You describe him as the 'Venerable' so I presume at some later stage he was promoted to Archdeacon. As such, he may have continued to serve 'his old boss' the Archbishop of Armagh whose diocese includes part of Tyrone. Alternatively he may have been attached to the Diocese of Clogher, or the Diocese of Raphoe, both of which also include large parts of Co. Tyrone.
A private subscription library in Belfast, the Linenhall Library, probably holds clergy lists and church histories. I will do a cursory search for you and if anything interesting turns up I will let you know.
Brian