Unlike fixed- or variable-rate image coding, progressive image coding (PIC) aims to compress various qualities of images into a single bitstream, increasing the versatility of bitstream utilization and providing high compression efficiency compared to simulcast compression. Research on neural network (NN)-based PIC is in its early stages, mainly focusing on applying varying quantization step sizes to the transformed latent representations in a hierarchical manner. These approaches are designed to compress only the progressively added information as the quality improves, considering that a wider quantization interval for lower-quality compression includes multiple narrower sub-intervals for higher-quality compression. However, the existing methods are based on handcrafted quantization hierarchies, resulting in sub-optimal compression efficiency. In this paper, we propose an NN-based progressive coding method that firstly utilizes learned quantization step sizes via learning for each quantization layer. We also incorporate selective compression with which only the essential representation components are compressed for each quantization layer. We demonstrate that our method achieves significantly higher coding efficiency than the existing approaches with decreased decoding time and reduced model size.