We consider the Landau–Lifshitz–Gilbert (LLG) equation, a nonlinear partial differential equation that models time-dependent micromagnetic phenomena. We present a fully discrete scheme combining first-order finite elements in space with a projection-free BDF2-type method of second order in time. Despite the nonlinearities of LLG, our scheme requires the solution of only one linear system at each time step. The unit-length constraint |m| = 1 is not enforced by normalization at each step, but is recovered in the limit. For a reduced model with only exchange energy and an applied external field, this is, to the best of our knowledge, the first linear higher-order-in-time integrator for LLG for which unconditional weak convergence to weak solutions and optimal a-priori error estimates of order O(h +k2) for sufficiently smooth solutions have both been established. Here, h and k denote the spatial mesh size and the time-step size, respectively. We then extend the scheme through an implicit-explicit (IMEX) strategy to incorporate additional energy contributions, including anisotropy, the nonlocal stray field, and the Dzyaloshinskii–Moriya interaction. The extension retains one linear solve per time step, unconditional weak convergence, and optimal a-priori error estimates. Numerical experiments illustrate the theoretical results.