For those wondering where the "Nan" came from within this surname the following speculation is offered by Ned Smith
There are 3 theories:
1st.. That the progenitor John Smith, Nan settled in Nantucket before moving to Hempstead
2nd.. That the word is a shortened form of some Welch dialect word meaning "hills"
3rd.. That the term is from an obsolete English word meaning "dwarf" (based on the same Latin root which gives us "nanosecond")

The fact that the Hempstead Town Records sometimes refer to John Smith, Nan, as "Little John Smith" seems to me to be nearly conclusive in favor of the 3rd theory.

One "Nan" researcher that has contributed to the making of this file is Phyllis.  Thank you to her for her efforts.