Nitrogen-vacancy (NV) centers can couple to confined phonons in diamond mechanical resonators via the effect of lattice strain on their energy levels. Access to the strong spin-phonon coupling regime with this system requires resonators with nanoscale dimensions in order to overcome the weak strain response of the NV ground-state spin sublevels. In this work, we incorporate photostable NVs in diamond cantilevers with lateral dimensions of a few hundred nanometers. Coupling of the NV ground-state spin to the mechanical mode is detected in electron spin resonance, and its temporal dynamics are measured via spin echo. Our small mechanical-mode volume leads to a 10 x -100x enhancement in the spin-phonon coupling strength over previous NV-strain coupling demonstrations. This is an important step towards strong spin-phonon coupling, which can enable phonon-mediated quantum-information processing and quantum metrology.