How do I place the liner so it doesn't overlap itself in places?
The best way to do this is to fold the very edges of the lining under and press them. Line the folded edges of the lining up with where it goes in the coat and sew it together using the slipstitch. That way the stitches are invisible and the lining and exterior fabric meet nicely without big seams.
This tutorial has some good photos:
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