Here’s something I did once. It’s kind of a round-about solution and the result wasn’t perfect, but it might be an improvement.
First I had a professional glossy print made of the small photo. Then I took that print and scanned it at extremely high resolution, probably something like 1200 dpi. That allowed me to sort of fake in the extra detail I needed to enlarge the photo.
Sorry, I replied without checking the exact size of the photo. I don’t think what I suggested what work on one that tiny.
Exactly what I was thinking. I just tried making it 500% bigger and it is out of focus.
I would even try printing it and then scanning it, just too see what happens.