Summary: Recently, G. E. Andrews et al. [Res. Number Theory 1, Paper No. 19, 25 p. (2015; Zbl 1386.11108)] introduced a new partition function \(p_{\omega}(n)\) that denotes the number of partitions of \(n\) in which each odd part is less than twice the smallest part. The generating function of \(p_{\omega}(n)\) is associated with the third-order mock theta function \(\omega (q)\). G. E. Andrews et al. [Ramanujan J. 43, No. 2, 347–357 (2017; Zbl 1421.11083)] proved three infinite families of congruences modulo 4 and 8 for \(p_{\omega}(n)\) and provided elementary proofs of congruences modulo 5 for \(p_{\omega}(n)\) which were first discovered by M. Waldherr [Proc. Am. Math. Soc. 139, No. 3, 865–879 (2011; Zbl 1279.11049)]. In this paper, we prove some new congruences modulo 5 and powers of 2 for \(p_{\omega}(n)\). In particular, we obtain some non-standard congruences for \(p_{\omega}(n)\). For example, we prove that for \(k\geq 0\), \( p_{\omega}\left( \frac{7\times 5^{2k+1}+1}{3}\right) \equiv (-1)^k \pmod 5 \) and \( p_\omega \left( \frac{2^{2k+7}+1}{3}\right) \equiv 1251 \times (-1)^k \pmod {2^{11}}\).