There is, unfortunately, no magic solution to this. I would be a little wary of a program that makes any great claims, as the only real way to get a sharp, big image is to start with one. That said, you can certainly make some enhancements, as long as you are good with "just okay".
For a free program, you might try GIMP and add the G'mic extension, using G'mic's enhancement tools to play around with no more than 25% enlargement to see if you can get something acceptable. There is a bit of a learning curve with GIMP, but it is every bit as powerful as Photoshop.
Otherwise, I'd bet Mr Tuttle has a few good suggestions.