Thanks Iamsam. Thought it would be easier to asnwer for a specific image rather than general situation yet I may have misunderstood the question so will provide a more general answer from my experience.
Hi Rwa
There are a couple factors that come in to play with having a good image in high definition when printed.
1) Do you have enough pixels in the original image? When viewing an image from 10 inch distance, the human eye can only resolve about 240 ppi. You can get decent pictures at about 150 ppi in my opinion. So here are the pixels you need in that range
4x6 print: 1200x1800 pixesl best quality 600 x 900 reasonable quality
8x10 print: 2400 x 3000 pixels best quality 1200 x 1500 reasonable quality
So you just need to determine how many pixels are in the image from your smart phone. More recent phones are really good quality and even older ones have reasonable number of pixels.
2) If the original image has issues such as the tone or color is off from ideal, most photo eidtors can hep with that. If the image is blurred, then less can be done in a photo editor yet there are some tricks to fool the eye to make it seem sharper. So improvement with these types of issues is a case by case depending on the image
Hope this helps
John Wheeler