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Nowadays, online screen sharing and remote cooperation are becoming ubiquitous. However, the screen content may be downsampled and compressed during transmission, while it may be displayed on large screens or the users would zoom in for detail observation at the receiver side. Therefore, developing a strong and effective screen content image (SCI) super-resolution (SR) method is demanded. We observe that the weight-sharing upsampler (such as deconvolution or pixel shuffle) could be harmful to sharp and thin edges in SCIs, and the fixed scale upsampler makes it inflexible to fit screens with various sizes. To solve this problem, we propose an implicit transformer network for continuous SCI SR (termed as ITSRN++). Specifically, we propose a modulation based transformer as the upsampler, which modulates the pixel features in discrete space via a periodic nonlinear function to generate features for continuous pixels. To enhance the extracted features, we further propose an enhanced transformer as the feature extraction backbone, where convolution and attention branches are utilized parallelly. Besides, we construct a large scale SCI2K dataset to facilitate the research on SCI SR. Experimental results on nine datasets demonstrate that the proposed method achieves state-of-the-art performance for SCI SR (outperforming SwinIR by 0.74 dB for x3 SR) and also works well for natural image SR. Our codes and dataset will be released upon the acceptance of this work.