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That meant that I had to be into it (the dancing) as well if I wanted to stay cool in her eyes. Maybe this is how people keep things fresh, by always doing new things. I finished my
second drink and had finally started to loosen up a bit.

Source:How To Disappear Completely

Background: She and I were having a party dancing to the music

Does (had finally started) happen after (I finished my second drink), right? I was confusing because past perfect happened prior to simple past. So I'm not quite sure. Thanks.
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Your interpretation is right. The grammar of the sentence is wrong.