We present an adaptive approximation scheme for jump-diffusion SDEs with discontinuous drift and (possibly) degenerate diffusion. This transformation-based doubly-adaptive quasi-Milstein scheme is the first scheme that has strong convergence rate $1$ in terms of the number of evaluations of the driving noise processes in $L^p$ for $p\in[1,\infty)$. To obtain our result, we prove that under slightly stronger assumptions, which are still weaker than those in the existing literature, a related doubly-adaptive quasi-Milstein scheme has convergence order $1$. This scheme is doubly-adaptive in the sense that it is jump-adapted, i.e. all jump times of the Poisson noise are grid points, and it includes an adaptive step-size strategy to account for the discontinuities of the drift.