High-temperature superconducting thin films of ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$ are examined by measuring the second-harmonic generated (SHG) signal from an incident 60-fs laser pulse of 2.0 eV (620 nm) in air at room temperature. As oxygen stoichiometry is varied (0.000.50), large changes in the SHG intensity are observed. The oxygen content of the films is independently determined by measuring (i) the c-axis lattice parameter derived from x-ray diffraction and (ii) the stoichiometry of the films determined by elastic backscattering spectrometry. We report a negative correlation of the SHG intensity with the c-axis parameter indicating that a known compression of the c axis leads to a larger SHG signal. An additional experiment in which the SHG intensity is monitored as a function of temperature (300--20 K) in vacuum is performed. A gradual increase of \ensuremath{\sim}50% in the SHG intensity is observed as the temperature is lowered, which can also be correlated to a decrease in the length of the c-axis parameter. Other experiments including rotation anisotropy and polarization of the incident and analyzer beams are also discussed.