  We prove infinitely many congruences modulo 3, 5, and powers of 2 for the
overpartition function $\bar{p}(n)$ and two smallest parts functions:
$\bar{\operatorname{spt1}}(n)$ for overpartitions and $\operatorname{M2spt}(n)$
for partitions without repeated odd parts. These resemble the Hecke-type
congruences found by Atkin for the partition function $p(n)$ in 1966 and Garvan
for the smallest parts function $\operatorname{spt}(n)$ in 2010. The proofs
depend on congruences between the generating functions for $\bar{p}(n)$,
$\bar{\operatorname{spt1}}(n),$ and $\operatorname{M2spt}(n)$ and eigenforms
for the half-integral weight Hecke operator $T(\ell^{2})$.
