The growing use of large language models has raised environmental and economic concerns about their intensity of resource usage during inference. Serving these models to each user requires substantial energy and water for cooling. Model compression techniques like quantization can shrink large language models and make them more resource efficient at the cost of potential performance degradation. Quantization methods compress model size through replacing their high-precision parameters by quantized values of lower precision. Among existing methods, the ApiQ method achieves superior accuracy preservation at minimal memory and time overhead. We investigate two ideas to extend performance in ultra-low-bit quantization beyond ApiQ's level. First, we look into combining existing quantization-aware training techniques with ApiQ's partial training. We show that this does not outperform the baseline ApiQ method with limited training data and frozen weights. This leads to two key insights: (1) The substantial representational capacity that is gained through full retraining is unlikely to be feasible through partial training. (2) This gain may depend on using a large and diverse dataset in quantization-aware training. Second, through a novel approach informed by the two insights, we propose an ultra-low-bit quantization method that builds upon ApiQ and extends its performance without the need for full retraining. This publicly available method relies on a saliency-aware regularization term that prioritizes preserving the most impactful parameters during quantization. Our experiments on LLaMA 7B and 13B benchmarks demonstrate that our method reduces the ApiQ's accuracy degradation by 10.85% and 7.54% respectively. A Python implementation of the proposed quantization method is publicly available on GitHub https://github.com/TokuyuSou/ULB-SAPR.