Let \(\overline{t}(n)\) denote the number of overpartitions of n where (i) the difference between two successive parts may be odd only if the larger part is overlined and (ii) if the smallest part is odd then it is overlined. In this work, we prove many infinite families of congruences modulo 16, 32 and 64 for \(\overline{t}(n)\) by using elementary generating function dissection techniques. For example, if m is a positive integer with \((m,3)=1\) and \(p\ne 3\) is a prime, then for \(n\ge 0\) , \(\begin{aligned} \overline{t}(m^2(72n+24)) \equiv \left\{ \begin{array}{ll} 8 \pmod {16}, & \hbox {if } n=r(3r+2) \text { for some integer } r; \\ 0 \pmod {16}, & \hbox {otherwise.} \end{array} \right. \end{aligned}\)
