Yes you can plant things after you have harvested your potatoes. Peas would probably be the best thing to plant if you want to harvest in the summer. Though, some lettuces are fairly quick to germinate and grow as well.
If memory serves the earliest potatoes are harvested mid summer. So, another idea would be to plant some winter crops, to get a head start on those (winter lettuces, chard, kohlrabi, broccoli and so on) you can plant them in little containers at the end of May that way, they will be up and ready for planting my the end of July. And you will be eating all winter long. I'm still eating my veggies from last winter and I have the new crop planted.