Summary: In this work, we consider the first-order dynamic equations \[ x^\Delta(t) + p(t)x(\tau(t)) =0,\; t\in[t_0,\infty)_{\mathbb{T}} \] where \(p\in C_{rd}([t_0, \infty)_{\mathbb{T}},\mathbb{R}^+)\), \(\tau\in C_{rd}([t_0, \infty)_{\mathbb{T}},\mathbb{T})\) and \(\tau(t)\leq t\), \(\lim_{t\rightarrow\infty}\tau(t)=\infty\). When the delay term \(\tau(t)\) is not necessarily monotone, we present a new sufficient condition for the oscillation of first-order delay dynamic equations on time scales.