It is well known that for a given set $S$ with well-founded order relation $R$, the lexicographic order that extends $R$ on tuples of $S$ is also well-founded.
Also, the multiset order on the multiset extension of $S$ is also well-founded.
Are there any other extensions to sets besides the lexicographic extension and the multiset extension that have a natural ordering that preserve well-foundedness?
I seem to remember reading that lexicographic and multiset extensions are the only extensions that preserve well-foundedness, but I cannot find the reference.
I am only asking about well-foundedness, i.e., not assuming the orderings are necessarily total.