High-dimensional quantum key distribution (QKD) offers secure communication, with secure key rates that surpass those achievable by QKD protocols utilizing two-dimensional encoding. However, existing high-dimensional QKD protocols require additional experimental resources, such as multiport interferometers and multiple detectors, thus raising the cost of practical high-dimensional systems and limiting their use. Here, we present and analyze a novel protocol for arbitrary-dimensional QKD, that requires only the hardware of a standard two-dimensional system. We provide security proofs against individual attacks and coherent attacks, setting an upper and lower bound on the secure key rates. Then, we test the new high-dimensional protocol in a standard two-dimensional QKD system over a 40 km fiber link. The new protocol yields a two-fold enhancement of the secure key rate compared to the standard two-dimensional coherent one-way protocol, without introducing any hardware modifications to the system. This work, therefore, holds great potential to enhance the performance of already deployed time-bin QKD systems through a software update alone. Furthermore, its applications extend across different encoding schemes of QKD qudits.