As the demand for deep learning grows, cost reduction through quantization has become essential for both training and inference. In 2022, the Open Compute Project (OCP) consortium standardized narrow precision formats for deep learning, called the microscaling (MX) format. The MX format is a hardware-friendly dynamic quantization scheme that effectively reduces the data size by sharing an 8-bit exponent across multiple operands. The MX format can be categorized into two types with their own strengths: (i) MXINT which focuses on a high precision consisting only of mantissa bits and (ii) MXFP which focuses on a wider dynamic range by allowing local exponent bits. In this work, we present a versatile MXFP format, called MX-SAFE (MXSF in short), that adaptively uses two modes, i.e., a wider mantissa mode (FP8 E2M5) and a subnormal FP mode (FP5 E3M2), to support both training and direct-cast inference. Furthermore, we propose a tile-based block design to increase hardware efficiency by reducing the burden of re-quantization process during the training with the MXSF format. Owing to the use of the proposed MXSF format, 0.05%/11.1% and 3.55%/3.57% improvements in accuracy, on average, for inference/full-training compared to MXFP8 E2M5 and MXFP8 E4M3 are observed, respectively. Moreover, we present a training-inference accelerator that supports the MXSF format and it achieves similar accuracy to the BF16 baseline while using 24.9% less total energy consumption.