We consider the tail distribution of the edge cover time of a specific non-Markov process, $δ$ once-reinforced random walk, on finite connected graphs, whose transition probability is proportional to weights of edges. Here the weights are $1$ on edges not traversed and $δ\in(0,\infty)$ otherwise. In detail, we show that its tail distribution decays exponentially, and obtain a phase transition of the exponential integrability of the edge cover time with critical exponent $α_c^1(δ)$, which has a variational representation and some interesting analytic properties including $α_c^1(0+)$ reflecting the graph structures.