I have never liked that statement; "soften the reins". To me, it mean more like, "get the horse to soften to the bit a little more". You can't do that by just opening your hand and dropping the contact.
To me , if someone says soften on the inside rein, I would very gently close my hand on the rein with my fingers so that it's the tiniest of "milking" motion to the rein. It's enough to get the horse's attention and ask him to give ever so slightly to that side, so the net result it that the HORSE creates some looseness there, and the result of that is that the hrose puts himself more on the outside rein, softer on the inside.
So , YOU don't soften that rein so much as you ask the horse to be soft on it, and you reward him by giving him a tiny release there.
Close your hands more around the rein. Being soft on the rein does not mean openning it so much that your fingers look all spidery out in every direction.