Bioinspiration can lead to exceptional mechanical properties in a number of biological materials as a result of their internal structure. In particular, the hierarchical arrangement of nano-to macro-components can bring to complex energy dissipation mechanisms and unprecedented resistance to crack growth. In this work, we propose to exploit this approach, combining in a multiscale composite structure carbon nanotubes with conventional carbon fibre reinforcements in a polyvinyl butyral matrix. We show that grafting the nanotubes onto the carbon microfibres improves their interface properties with the matrix considerably, effectively doubling their apparent strength. At the same time, the addition of nanotubes to microfibre reinforcements helps to improve the composite toughness, reaching more than twice the value for the conventional, non-hierarchically reinforced composite. Numerical simulations and fracture mechanics considerations are also provided to interpret the results.