Understanding the cause of instability of viscoelastic liquid jets is of fundamental importance to realise practical applications such as ink-jet printing, fuel injection spraying, and fibre spinning. For common viscoelastic fluids, the elastic stresses caused by the stretching of entangled polymeric chains at the nozzle can persist along the jet for a long downstream distance far away from the exit. Unrelaxed elastic tension has been regarded as a factor responsible for the delay in the breakup of viscoelastic jets. The present study performs a complete linear stability analysis to obtain deeper insights into the response of unrelaxed elastic tension to the onset of three-dimensional instability in viscoelastic jets. Results show that the elastic effect is multidimensional. In the absence of the unrelaxed tension, the elastic force, characterised by a Deborah number, has a slightly destabilising influence on both axisymmetric and non-axisymmetric disturbances. However, when the unrelaxed tension arises, the elastic force begins to suppress the instabilities driven by surface tension and the wind-induced effect to a great extent. In particular, once the tension exceeds a threshold, some novel oscillating non-axisymmetric modes emerge because of stimulation by the relaxation of the elastic energy, causing a variety of asymmetrical deformations.