Transformer-based Super-Resolution (SR) models have recently advanced image reconstruction quality, yet challenges remain due to computational complexity and an over-reliance on large patch sizes, which constrain fine-grained detail enhancement. In this work, we propose TaylorIR to address these limitations by utilizing a patch size of 1x1, enabling pixel-level processing in any transformer-based SR model. To address the significant computational demands under the traditional self-attention mechanism, we employ the TaylorShift attention mechanism, a memory-efficient alternative based on Taylor series expansion, achieving full token-to-token interactions with linear complexity. Experimental results demonstrate that our approach achieves new state-of-the-art SR performance while reducing memory consumption by up to 60% compared to traditional self-attention-based transformers.