Thanks for your interest, moonmom. It was good of you to look it up and post your findings.
The Henry Coe I'm researching moved to South Elmham from elsewhere. His marriage, as a widower, to Mary Elwin, in South Elmham St. Margaret in 1776, and the burial of a Mary Coe in 1774, are the first references of the name Coe in any South Elmham parish, or even surrounding parishes, that I've come across. Henry Coe must, therefore, have moved there from quite a number of parishes away, but just how far away, I have not been able to discover.
This is position I've reached, so I am still looking for a helping hand from anyone who may have come across a baptism of any Henry Coe at the right date (circa 1747), even if he was baptised a long way from South Elmham itself. It might give a lead which I can follow up with some more research.