Controller Area Network (CAN) is a widely used protocol that allows communication among Electronic Control Units (ECUs) in automotive electronics. It was extended to CAN-FD (CAN with Flexible Data-rate) to meet the increasing demand for bandwidth utilization caused by the growing number of features in modern automobiles. The signal-to-frame packing problem has been studied in literature for both CAN and CAN-FD. In this work, we propose and formulate, for the first time, the signal offset assignment problem (SOAP) in a frame in order to improve the bus bandwidth utilization. We prove that SOAP is NP-complete. We propose a general approximation framework (GAF) for SOAP which can use any approximation algorithm for the makespan minimization problem (MMP) in multiprocessor systems. We derive the performance guarantee provided by GAF as a function of the performance guarantee of the approximation algorithm for MMP and the number of signal periods in the frame. We demonstrate the efficacy of our approach through experiments using three different algorithms (two approximation algorithms and an integer linear programming formulation) for MMP in GAF. Our results indicate that by using offsets for signals in frame packing schemes, one can achieve about 10.54% improvement in bandwidth utilization (on a single bus) in CAN-FD systems.