A new embedded low-regularity integrator is proposed for the quadratic nonlinear Schrödinger equation on the one-dimensional torus. Second-order convergence in $H^\gamma$ is proved for solutions in $C([0, T]; H^\gamma)$ with $\gamma > \frac{3}{2},$ i.e., no additional regularity in the solution is required. The proposed method is fully explicit and can be computed by the fast Fourier transform with $\mathcal{(O} log N)$ operations at every time level, where $N$ denotes the degrees of freedom in the spatial discretization. The method extends the first-order convergent low-regularity integrator in [14] to second-order time discretization in the case $\gamma >\frac{3}{2}$ without requiring additional regularity of the solution. Numerical experiments are presented to support the theoretical analysis by illustrating the convergence of the proposed method.