Some people will tell you that heating till it is Cherry Red and then letting it Air Cool (Annealing) can held. But 316L's has a relatively Small Hardness Range due to it's Low Carbon content. And I don't think Kanthal A1 is or can get very hard and Springy.
As you noticed, Keeping everything under Tension is a Good Call.
Check out this Dude and how he keeps Constant Tension on the Core as well as the Wrap.
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He also has a Dozen or so "Advanced" video's for all kinds of Wire Porn.