I know that by Tennenbaum's Theorem, non-standard models of arithmetic "don't know" which of their elements form a standard model of arithmetic. However, often facts that are opaque to the model from an "internal" perspective can be recognized "from the outside". (After all, if this weren't possible we wouldn't know that these models were non-standard.) For example, countable models of set theory can be seen to be "non-standard", in a sense, because they classify certain sets as uncountable when "from the outside" we can see this is simply a weird side-effect of that model lacking an injective function from these sets into their natural numbers.
So, suppose that from an external perspective we know which substructure of a non-standard model of arithmetic contains the standard numbers (even though the model can't identify this substructure from its perspective) -- perhaps we started with a standard model and constructed the non-standard model from it.
Could the construction of the non-standard model add to the properties of the standard numbers we started with? Or does the construction of the non-standard model always add completely disconnected structure to the initial structure, such that the properties of the initial structure remain completely unchanged?