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 Rea for this instability is an order of magnitude smaller than the critical Rea predicted before for the helical instability (where Rea = rva/v, r is the distance from the jet source, va is the jet maximum velocity at a given r, and v 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 Res (circulation to viscosity ratio) exceeding 3, the critical Rea for the single-helix counter-rotating mode becomes smaller than those for axisymmetric and multi-helix modes. Since the critical Res 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.