The near-yrast structure of the near-magic, odd-odd nucleus, ${}_{39}^{88}$Y${}_{49}$, has been studied into the high-spin regime. Investigations were performed at the Wright Nuclear Structure Laboratory, Yale University, using the ${}^{74}$Ge(${}^{18}$O,$p3n$) and ${}^{76}$Ge(${}^{18}$O,$p5n$) fusion-evaporation reactions at beam energies of 60 and 90 MeV, respectively. Gamma-ray energy coincidence analyses using both double (${\ensuremath{\gamma}}^{2}$) and triple (${\ensuremath{\gamma}}^{3}$) fold coincidences, together with angular correlation measurements, have been used to extend the previously reported level scheme to an excitation energy of 8.6 MeV and a spin and parity of 19${}^{(\ensuremath{-})}$. The presented level scheme is compared with predictions of a truncated valence space shell-model calculation, which assumes an inert ${}^{56}$Ni core with proton and neutron excitations allowed within the ${f}_{5/2}$, ${p}_{3/2}$, ${p}_{1/2}$, and ${g}_{9/2}$ single-particle states. The shell-model calculations show a reasonable comparison with the experimental data for the yrast, positive-parity states up to spin 18 $\ensuremath{\hbar}$, with larger variations evident for negative-parity states with spins greater than 16 $\ensuremath{\hbar}$. In spite of a significant increase in angular momentum input associated with the thin target ${}^{76}$Ge(${}^{18}$O,$p5n$) reaction channel, as compared to the backed target data using the ${}^{74}$Ge target, no additional discrete states were identified in the former data set, suggesting that the level scheme for this nucleus fragments significantly above the observed states, possibly indicating cross-shell excitations becoming dominant for $Ig$19 $\ensuremath{\hbar}$.