  In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects
which they called \emph{partitions with designated summands}. These are built
by taking unrestricted integer partitions and designating exactly one of each
occurrence of a part. In that same work, Andrews, Lewis, and Lovejoy also
studied such partitions wherein all parts must be odd, and they denoted the
number of such partitions of size $n$ by the function $PDO(n)$. Since then,
numerous authors have proven a variety of divisibility properties satisfied by
$PDO(n)$. Recently, the second author proved the following internal congruences
satisfied by $PDO(n)$: For all $n\geq 0$, \begin{align*} PDO(4n) &\equiv PDO(n)
\pmod{4},\\ PDO(16n) &\equiv PDO(4n) \pmod{8}. \end{align*} In this work, we
significantly extend these internal congruence results by proving the following
new infinite family of congruences: For all $k\geq 0$ and all $n\geq 0$,
$$PDO(2^{2k+3}n) \equiv PDO(2^{2k+1}n) \pmod{2^{2k+3}}.$$ We utilize several
classical tools to prove this family, including generating function dissections
via the unitizing operator of degree two, various modular relations and
recurrences involving a Hauptmodul on the classical modular curve $X_0(6)$, and
an induction argument which provides the final step in proving the necessary
divisibilities. It is notable that the construction of each $2$-dissection
slice of our generating function bears an entirely different nature to those
studied in the past literature.
