What are the practical differences between 1/2" and 3/4" pipe between a water softener and water heater?
Provided the line coming into your house from the city main is 3/4&quot; and inside your house the line changes to 1/2&quot; there would be a significant increase in flow rate and possible pressure when water demand is higher in your house.
Run a new 3/4&quot; line to the softener as described.  They could require the extra flow rate when it regenerates (which could use ~25 gallons/~0.11 m3) when it does this.