This is one of those threads where I did not want to be the first to offer bad advice! Which is not to say the above is bad.

You are at point where you don't want to do anything that will kill a saddle that may already be dead. If it is as soft as you say, there is probably not much strength left in it. So tension may help, or it may just rip it apart.

Either way, I expect this piece of leather is going to tear into two pieces in another couple hundred miles and there ain't nothing you can do about it. Either hang it on the wall as art, and preserve it as is; or ride it and maybe kill it.

If you do kill it, please send me the remains That said:

I suspect the best you can do is cut a piece of firm foam to fit as neatly as possible between the leather and the rails, and stuff it in there. This will support the leather from underneath. And it can do no harm.

Adding tension would be a little more radical. It may, or may not, work.

Even more radical would be to melt beeswax into the leather. That will harden it for sure, but I question whether the leather can take that heat.