I have my potatoes planted in "Potato Bags" that I made. Basically, it is a 18' diameter bag of landscape fabric that I put into a cylinder of machine fabric to keep the gophers out.
One bag got planted at the beginning of June, the plants in that bag are turning yellow and dropping leaves. I am going to dump it tomarrow night to see if I got any potatoes. The other two bags were planted at the beginning of July, they are still green and growing. They get to stay for another couple of weeks, until the cold gets them.
I believe you should wait until the plants start to die back to get the most growth of potatoes.
However, you can dig at any time and as long as you don't disturb the roots, they will continue to grow.
In WI, I would carefully dig around the plants to pick the small reds during the late summer and fall. Then, when the plants died back, I would do the "big dig" and harvest any that were left. Here in WY, our soil is so bad that I had to resort to mixing my own soil with bagged stuff from Lowes and planting in bags. I actually like this better. I add soil to the bag as the plants grow, I started with 6" of soil and now have over a foot.