This study, employing a numerical approximation, computationally describes 2D melt fracture as elastic instability in the flow along and outside a straight channel. In the preceding research (Kwon, 2018, Numerical modeling of two-dimensional melt fracture instability in viscoelastic flow, J. Fluid Mech. 855, 595-615) several types of unique instability and corresponding bifurcations such as subcritical and chaotic transitions have been illustrated with possible mechanism presumed. However, the 1st bifurcation from stable steady to unstable periodic state could not be accurately characterized even though its existence was proven evident. The analysis herein aims at verification of this 1st transition to temporally (and also spatially) periodic instability, utilizing the same numerical technique with attentive control of flow condition. As a result of scrutinizing the solutions, the steady elastic flow described by the Leonov rheological model passes through supercritical Hopf bifurcation at the Deborah number of 10.42 and then transforms to the state of the 1st weak periodic instability. It has also been confirmed that near this bifurcation point it takes extremely long to completely develop into either steady state (in the stable case) or periodic instability, which obstructed immediate characterization of the transition in the previous work.