By training over large-scale datasets, zero-shot monocular depth estimation (MDE) methods show robust performance in the wild but often suffer from insufficient detail. Although recent diffusion-based MDE approaches exhibit a superior ability to extract details, they struggle in geometrically complex scenes that challenge their geometry prior, trained on less diverse 3D data. To leverage the complementary merits of both worlds, we propose BetterDepth to achieve geometrically correct affine-invariant MDE while capturing fine details. Specifically, BetterDepth is a conditional diffusion-based refiner that takes the prediction from pre-trained MDE models as depth conditioning, in which the global depth layout is well-captured, and iteratively refines details based on the input image. For the training of such a refiner, we propose global pre-alignment and local patch masking methods to ensure BetterDepth remains faithful to the depth conditioning while learning to add fine-grained scene details. With efficient training on small-scale synthetic datasets, BetterDepth achieves state-of-the-art zero-shot MDE performance on diverse public datasets and on in-the-wild scenes. Moreover, BetterDepth can improve the performance of other MDE models in a plug-and-play manner without further re-training.