I definitely hear the country influence in That's All. The Alison Krauss and Vince Gill cover makes it even more obvious:
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Agree that The Roof Is Leaking also has country in it. Me and Virgil has a bit too.
People haven't mentioned soul and R&B, maybe because there are lots of examples: Misunderstanding, No Reply At All, Sledgehammer, etc. Peter also used to cover Marvin Gaye in his early solo tours.