  In 2022, Broudy and Lovejoy extensively studied the function $S(n)$ which
counts the number of overpartitions of \emph{Schur-type}. In particular, they
proved a number of congruences satisfied by $S(n)$ modulo $2$, $4$, and $5$. In
this work, we extend their list of arithmetic properties satisfied by $S(n)$ by
focusing on moduli which are small powers of 2. In particular, we prove the
following infinite family of Ramanujan-like congruences: For all $\alpha\geq 0$
and $n\geq 0$, $$
S\left(2^{5+2\alpha}n+\left(2^{5+2\alpha}-\frac{2^{2+2\alpha}-1}{3}\right)\right)\equiv
0 \pmod{16}. $$ All of the proof techniques used herein are elementary, relying
on classical $q$-series identities and generating function manipulations as
well as the parameterization work popularized by Alaca, Alaca, and Williams.
