Parameter-efficient fine-tuning (PEFT) techniques have unlocked the potential to cheaply and easily specialize large pretrained models. However, the most prominent approaches, like low-rank adapters (LoRA) depend on heuristics or rules-of-thumb for their architectural choices---potentially limiting their performance for new models and architectures. This limitation suggests that techniques from neural architecture search could be used to obtain optimal adapter architectures, but these are often expensive and difficult to implement. We address this challenge with Monarch Rectangular Fine-tuning (MoRe), a simple framework to search over adapter architectures that relies on the Monarch matrix class. Theoretically, we show that MoRe is more expressive than LoRA. Empirically, our approach is more parameter-efficient and performant than state-of-the-art PEFTs on a range of tasks and models, with as few as 5% of LoRA's parameters.