Use cold-stored potato tubers, available from specialist seed merchants in July and August. These are seed potatoes from late winter that have been held back ready for summer planting.
First and second early varieties such as ‘Charlotte’, ‘Nicola’ and ‘Maris Peer’ are recommended. As these will go straight into warm soil, they do not need to be chitted prior to planting.
You can hold back late-winter-bought tubers yourself by keeping them in the fridge or leaving them in a cool, light place right through the spring and early summer and allowing them to develop long (and rather fragile) sprouts. These will need to checked regularly for aphids.
Potatoes harvested in summer and then replanted will not produce tubers for winter as they go through a long period of dormancy after harvest.