Double-layer targets composed of near-critical-density carbon nanotube foams (CNFs) and solid foils have shown their advantages in laser-driven ion acceleration under high relativistic intensity. Here, we report the experimental and numerical results on the laser-accelerated proton beams from such targets under moderate relativistic intensities [Formula: see text]. 40-TW femtosecond laser pulses were used to irradiate CNF-based double-layer targets. Compared to single-layer targets, significant enhancements on the cutoff energy and numbers of ions were observed. It was found that the CNF layer also leads to a larger divergence angle and a more homogeneous spatial distribution profile of the proton beam. Particle-in-cell simulations reveal the reason for the enhanced proton acceleration. It is found that the lateral electric field and the strong magnetic field built by the directly accelerated electrons from the CNF layer contribute to the enlarged divergence angle.