First things first:
There's no free lunch!
Best option is to try different interpolation methods in Photoshop, as well as techniques involving up-sizing incrementally instead of "all-at-once"....and there's also Genuine Fractals software that is designed exactly for your purpose although I've never used it.
Keep in mind that, if you're doing final imaging/printing of an image, it will get interpolated up the resolution required for printing (printer resolution) anyway. Whether the printing pipeline does a better or worse job of this than doing it manually in Photoshop or elsewhere is up to you.
Regards,
Terry