In the standard SIR model, infected vertices infect their neighbors at rate $λ$ independently across each edge. They also recover at rate $γ$. In this work we consider the SIR-$ω$ model where the graph structure itself co-evolves with the SIR dynamics. Specifically, $S-I$ connections are broken at rate $ω$. Then, with probability $α$, $S$ rewires this edge to another uniformly chosen vertex; and with probability $1-α$, this edge is simply dropped. When $α=1$ the SIR-$ω$ model becomes the evoSIR model. Jiang et al. proved in \cite{DOMath} that the probability of an outbreak in the evoSIR model converges to 0 as $λ$ approaches the critical infection rate $λ_c$. On the other hand, numerical experiments in \cite{DOMath} revealed that, as $λ\to λ_c$, (conditionally on an outbreak) the fraction of infected vertices may not converge to 0, which is referred to as a discontinuous phase transition. In \cite{BB} Ball and Britton give two (non-matching) conditions for continuous and discontinuous phase transitions for the fraction of infected vertices in the SIR-$ω$ model. In this work, we obtain a necessary and sufficient condition for the emergence of a discontinuous phase transition of the final epidemic size of the SIR-$ω$ model on \ER\, graphs, thus closing the gap between these two conditions.