Thanks for the information Christopher,
From
http://www.wikipedia.org it looks like Clogher is in the Dungannon district and Eskra is in the Omagh district.
To update my original post, I think the boundaries/names of the districts were not changed starting 1956, instead, the indexes listed the more specific registration district instead of the larger Superintendent Registrar's District.
For example, the Clogher Superintendent Registrar's district contains the towns/villages of Clogher, Augher, Fivemiletown, Aughnacloy, Ballygawley, Eskra in County Tyrone and a small area of County Monaghan. Prior to 1956, births etc in these towns and surrounding rural areas would just be listed as "Clogher" in the index, but when you examined the actual record it would list both Clogher and the more specific district name such as "Ballygawley". But there was no way to search for births occurring in Ballygawley only.
"Clogher" was composed of four districts: Aughnacloy, Ballygawley, Clogher and Fivemiletown, and after 1956 births and deaths are indexed under these 4 names. Likewise Omagh was composed of subdistricts: Gortin, Fintona, Omagh 1, Omagh 2 etc. and these are the names that appear in the indexes after 1956.
So it's not really a problem when you are familiar with the districts in an area, it actually helps your search be more specific. The reason I didn't immediately notice how the new indexing worked is that my surnames of interest are rare enough that I didn't find any listings for them in these districts.
But for areas that you're not familiar with, you would need to find out the names of the districts in the area, and the district councils would be a good place to start.
Bernie