The move-acceptance hyper-heuristic was recently shown to be able to leave local optima with astonishing efficiency (Lissovoi et al., Artificial Intelligence (2023)). In this work, we propose two modifications to this algorithm that demonstrate impressive performances on a large class of benchmarks including the classic CLIFF_d and JUMP_m function classes. (i) Instead of randomly choosing between the only-improving and any-move acceptance operator, we take this choice via a simple two-state Markov chain. This modification alone reduces the runtime on JUMP_m functions with gap parameter m from ?(n²ᵐ⁻¹) to O(nᵐ⁺¹). (ii) We then replace the all-moves acceptance operators with the operator that only accepts worsenings. Such a, counter-intuitive, operator has not been used before in the literature. However, our proofs show that our only-worsening operator can greatly help in leaving local optima, reducing, e.g., the runtime on Jump functions to O(n³ log n) independent of the gap size. In general, we prove a remarkably good runtime of O(nᵏ⁺¹ log n) for our Markov move-acceptance hyper-heuristic on all members of a new benchmark class SEQOPT_k, which contains a large number of functions having k successive local optima, and which contains the commonly studied JUMP_m and CLIFF_d functions for k=2.