We investigate subdominant order parameters stabilizing at low temperatures in nanoscale high-T(c) cuprate islands, motivated by the recent observation of a fully gapped state in nanosized YBa(2)Cu(3)O(7-δ) [D. Gustafsson et al., Nature Nanotech. 8, 25 (2013)]. Using complementary quasiclassical and tight-binding Bogoliubov-de Gennes methods, we show on distinctly different properties dependent on the symmetry being d(x(2)-y(2))+is or d(x(2)-y(2))+id(xy). We find that a surface-induced d(x(2)-y(2))+is phase creates a global spectroscopic gap which increases with an applied magnetic field, consistent with experimental observation.