  Odd $K$-theory has the interesting property that it admits an infinite number
of inequivalent differential refinements. In this paper we provide a bundle
theoretic model for odd differential $K$-theory using the caloron
correspondence and prove that this refinement is unique up to a unique natural
isomorphism. We characterise the odd Chern character and its transgression form
in terms of a connection and Higgs field and discuss some applications. Our
model can be seen as the odd counterpart to the Simons-Sullivan construction of
even differential $K$-theory. We use this model to prove a conjecture of
Tradler-Wilson-Zeinalian regarding a related differential extension of odd
$K$-theory
