The next thing to do is to create more warmth and solidity for the new lining of your vintage fur coat. You do it with sewing decorative padding lining.
With my BlueBerryPie & VanillaCream PotHolder -tutorial you can learn how to sew exact square decorative padding.
This decorative sewing is actually able to be made by decorative random padding too. You can take a look at this method at According to Tone Finnager where I used random sewing method when sewing the basket to store the tissues.
Here I use the optional extra equipment which usually comes with the sewing machine to create decorative padding. This method is not so exact as you learnt before with the pot holder tutorial. But this is exact enough for the need we have here. I promise, no one will notice you did it a bit easier way.
At the end it doesn't matter which method you use to fasten the cotton wool fabric to lining fabric as long as you fasten it. Fastening creates more air pockets which store even more warmth in. Creating the pockets is the point, not the way of doing it. Whilst you create the warm air pockets you happen to make the new lining more solidity. It looks even more luxurious; despite we aim to recycling.
So, I sew 10 x 10 cm squares.
The back side looks like this...
And now, when you have done all your useful decorative sewing, only now you sew all your needed folds to the pattern pieces.