Using a fully-implicit high-resolution two-layer quasi-geostrophic model combined with pseudo-arclength continuation methods, we perform a bifurcation analysis of double-gyre ocean flows to study their initial oscillatory instabilities. In this model, both wind- and thermally-forced flows can be represented. We demonstrate that on the branch of anti-symmetric steady-state solutions the ratio, Ω, of the flow advective speed to the long internal Rossby wave speed determines the type of oscillatory modes to first become unstable. This is the same nondimensional parameter that controls the shape of the geostrophic contours in the linear limit of the circulation. For large values of Ω, the first Hopf bifurcations correspond to the classical baroclinic modes with inter-monthly time periods arising from shear instability of the flow. For small values of Ω, the first Hopf bifurcations correspond instead to barotropic Rossby modes with shorter, monthly periods arising from mixed barotropic-baroclinic instability of the flow. By considering both a wind-forced and a thermally-forced ocean, we show that this is a robust feature that does not depend on the type of forcing driving the circulation.