BLAS is a fundamental building block of advanced linear algebra libraries and many modern scientific computing applications. GPUs are known for their strong arithmetic computing capabilities and are highly suited for BLAS operations. However, porting code to GPUs often requires significant effort, especially for large, complex codes or legacy codes, even for BLAS-heavy applications. While various tools exist to automatically offload BLAS to GPUs, they are often impractical due to the high costs associated with mandatory data transfers. The advent of unified memory architectures in recent GPU designs, such as the NVIDIA Grace-Hopper, allows cache-coherent memory access across all types of memory for both CPU and GPU, potentially eliminating the bottlenecks faced in conventional architectures. This breakthrough paves the way for innovative application developments and porting strategies. Building on our preliminary work demonstrating the potential of automatic *gemm offload, this paper extends the framework to all level-3 BLAS operations and introduces SCILIB-Accel, a novel tool for automatic BLAS offload. SCILIB-Accel leverages the memory coherency in Grace-Hopper and introduces a Device First-Use data movement policy inspired by the OpenMP First-Touch approach in multi-socket CPU programming, minimizing CPU-GPU data transfers for typical scientific computing codes. Additionally, utilizing dynamic binary instrumentation, the tool intercepts BLAS symbols directly from a CPU binary, requiring no code modifications or recompilation. SCILIB-Accel has been evaluated using multiple quantum physics codes on up to a few hundred GPU nodes, yielding promising speedups. Notably, for the LSMS method in the MuST suite, a 3x speedup was achieved on Grace-Hopper compared to Grace-Grace.