Potato tubers are not actually roots. They are modified stems. So the surest way to force more potatoes is to “hill” them. In the commercial fields this is done with a huge tractor raking soil from in between planting rows and piling it up on the plants. You essentially bury the plants stem as it grows taller. Then the buds on the stem will push out stolons (horizontal underground stems.) these will terminate in tubers, aka: potatoes!

Source: did potato disease research for my PhD.

Additional edit: loose/sandy soil is critical. Too dense of soil and your tubers can’t expand well.

Don’t plant them too close to each other. It doesn’t work.