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As long it's not losing too much quality then it's okay. I would like to see the resultMuch can be done with Photoshop, but it's not magic. The image can be enlarged, but it's not gonna look good.

How about this? I've converted the previous picture from JPG to PNG. Will that help?I can run it through a couple tools yet do you have a version of the image that is not at such a high compression level of JPEG? To show the issue I have extracted in clockwise fashion the Hue component, luminosity component, and saturation component. Hue and Sat are already quite off and lots of compression artifacts.
Will do best if you don' have a better original.
John Wheeler

Ah, my bad. Indeed, I only have the previous picture one as the original sourceNo, all the aritfacts just carry over to the PNG. You would have to start from a better original. You may not have that I understand. Also, if you wanted a 6X on both dimensions, that would be 3840x2556. I will give it a go as others have already