We present an inflationary model preceded by a bounce in a metric theory à la $f(R)$, where $R$ is the scalar curvature of the space-time. The model is asymptotically de Sitter such that the gravitational action tends asymptotically to an Einstein-Hilbert action with an effective cosmological constant; therefore, modified gravity affects only the early stages of the Universe. We then analyze the spectrum of the gravitational waves through the method of the Bogoliubov coefficients by two means: taking into account the gravitational perturbations due to the modified gravitational action in the $f(R)$ setup and simply considering those perturbations inherent to the standard Einstein-Hilbert action. We show that there are distinctive (oscillatory) signals on the spectrum for very low frequencies; i.e., corresponding to modes that are currently entering the horizon.