I know this thread is kind of old but I thought I'd reply.
I'm descended from Thomas Jefcoate who emigrated to New Zealand in 1865. Certainly in most newspaper articles & on official documents in NZ, this line of the family are all listed as "Jefcoate". However, I think there might also be Jeffcotes in NZ but they are from another line of the family. And then, of course, there are the seemingly endless variations in the UK. It's astonishing how much a name can change over the years.
I'd be interested in hearing more about the family back in England or further afield. I'm avail via firstname.lastname@example.org