Quasi-analytic wave-front sets of distributions which correspond to the Gevrey sequence $p!^s$, $s\in[1/2,1)$ are defined and investigated. The propagation of singularities are deduced by considering sequences of Gaussian windowed short-time Fourier transforms of distributions which are modifications of the original distributions by suitable restriction-extension techniques. Basic micro-local properties of the new wave-fronts are thereafter established.