We study strong approximation of $d$-dimensional stochastic differential equations (SDEs) with a discontinuous drift coefficient driven by a $d$-dimensional Brownian motion $W$. More precisely, we essentially assume that the drift coefficient $μ$ is piecewise Lipschitz continuous with an exceptional set $Θ\subset \mathbb{R}^d$ that is an orientable $C^5$-hypersurface of positive reach, the diffusion coefficient $σ$ is assumed to be Lipschitz continuous and, in a neighborhood of $Θ$, both coefficients are bounded and $σ$ is non-degenerate. Furthermore, both $μ$ and $σ$ are assumed to be $C^{1}$ with intrinsic Lipschitz continuous derivative on $\mathbb{R}^{d}\setminus Θ$. We introduce, for the first time in literature, a Milstein-type method which can be used to approximate SDEs of this type for general $d \in \mathbb{N}$ and prove that this Milstein-type scheme achieves an $L_{p}$-error rate of order at least $3/4-$ in terms of the number of steps. This method depends, in addition to evaluations of $W$ on a fixed grid, also on iterated integrals w.r.t. components of $W$, which can in general not be represented as functionals of $W$ evaluated at finitely many time points. We additionally prove that our suggested Milstein-type method is only dependent on evaluations of $W$ on a finite, fixed grid if $σ$ is additionally commutative. To obtain our main result we prove that a quasi-Milstein scheme achieves an $L_{p}$-error rate of order at least $3/4-$ in our setting if $μ$ is additionally continuous, which is of interest in itself.