Best way to convert 5m resolution height map image to 1m resolution??

I’ve created a height map image based on real world elevation data from a 5m sample grid.
When I import it into Unreal and scale using an X/Y factor of 500 to achieve the real world dimensions, each vertex is 5m apart which it too coarse to sculpt new terrain features.

I assume I need to convert my 5m sample height map to a 1m sample height map (i.e. 1 pixel for every meter instead of 1 pixel for every 5 meters) but how can I do that without creating ugly steps? Any tips please on scaling up height maps? Perhaps there is a piece of software that makes a good job of this?