Large language models (LLMs) have demonstrated remarkable capabilities in various kinds of tasks, while the billion or even trillion parameters bring storage and efficiency bottlenecks for inference. Quantization, particularly floating-point quantization, is known to be capable of speeding up LLM inference by reducing memory footprint and data movement during the inference process. For the first time, we advance the floating-point quantization exploration from integer bitwidths to non-integer bit-widths, namely AMS-Quant, to further approach the quantization sweet spot. AMS-Quant incorporates two novel techniques to put it into effect: (1) it proposes Mantissa-bit Sharing, which groups k quantized weights and lets them share the least significant mantissa bit, allowing us to further approach the minimum quantization bit-width without accuracy loss. (2) It introduces Adaptive Searching, which employs an offline optimization strategy to minimize the accuracy degradation introduced by sharing. Moreover, AMS-Quant is also prototyped as efficient CUDA Linear kernels, which translates memory savings into wall-clock latency reduction by reducing memory access. Extensive experiments on large-scale datasets and models show that AMS-Quant can quantize the model to FP-5.33-e2m3 and FP4.25-e2m2, and significantly speed up the LLM decoding over FP16 inference (2.8x and 3.2x), with negligible accuracy loss.