A non-parallel analysis of time-oscillatory instability of conical jets reveals important features not found in prior studies. Flow deceleration significantly enhances the shear-layer instability for both swirl-free and swirling jets. In swirl-free jets, flow deceleration causes the axisymmetric instability (absent in the parallel approximation). The critical Reynolds number $\hbox{\it Re}_{a}$ for this instability is an order of magnitude smaller than the critical $\hbox{\it Re}_{a}$ predicted before for the helical instability (where $\hbox{\it Re}_{a}= rv_{a}/\nu, r$ is the distance from the jet source, $v_a$ is the jet maximum velocity at a given $r$, and $\nu$ is the viscosity). Swirl, intensifying the divergence of streamlines, induces an additional, divergent instability (which occurs even in shear-free flows). For the swirl Reynolds number $\hbox{\it Re}_s$ (circulation to viscosity ratio) exceeding 3, the critical $\hbox{\it Re}_a$ for the single-helix counter-rotating mode becomes smaller than those for axisymmetric and multi-helix modes. Since the critical $\hbox{\it Re}_s$ is less than 10 for the near-axis jets, the boundary-layer approximation (used before) is invalid, as is Long's Type II boundary-layer solution (whose stability has been extensively studied). Thus, the non-parallel character of jets strongly affects their stability. Our results, obtained in a far-field approximation allowing reduction of the linear stability problem to ordinary differential equations, are more valid for short wavelengths.