Non-linear inverse Compton scattering (NICS) is of significance in laser-plasma physics and for application-relevant laser-driven photon sources. Given this interest, we investigated this synchrotron-like photon emission in a promising configuration achieved when an ultra-intense laser pulse interacts with a double-layer target (DLT). Numerical simulations with two-dimensional particle-in-cell codes and analytical estimates are used for this purpose. The properties of NICS are shown to be governed by the processes characterizing laser interaction with the near-critical and solid layers composing the DLT. In particular, electron acceleration, laser focusing in the low-density layer, and pulse reflection on the solid layer determine the radiated power, the emitted spectrum, and the angular properties of emitted photons. Analytical estimates, supported by simulations, show that quantum effects are relevant at laser intensities as small as ∼ 10$^{21}$ W/cm$^2$ Target and laser parameters affect the NICS competition with bremsstrahlung and the conversion efficiency and average energy of emitted photons. Therefore, DLT properties could be exploited to tune and enhance photon emission in experiments and future applications.