If you start with piano wire, it is already spring tempered. No further heat treat necessary.
When the spring is wound, the wire is overstretched, i.e coiled too tightly. When the tension is released, it will spring back, increasing the coil diameter somewhat. Typically, you start with a slightly smaller mandrel so the relaxed diameter meets your requirement.
A refinement is to add a wire guide that fits up close to the mandrel. This prevents the already wound coils from unwinding, requiring less tension on the wire and having less spring-back. This is especially useful when winding heavier gauge springs.
One of the coolest ways to make a spring is to push the wire against an angled block. The wire will form a coil, the diameter determined by the angle of the block. This works great for making conical springs, something that I struggled with winding over a mandrel.