Given a well ordering of a set $A$ we can define a total order $A^A$ in an obvious way (for $f \neq g$ find the least $i$ such that $f(i) \neq g(i)$ and define $f < g$ if $f(i) < g(i)$)

Does the inverse direction work? Does a total order on the powerset of $A$ give rise to a well ordering of $A$? (without choice, of course, for otherwise the result is obvious)