Sie waren hinter ihr her.

English Translation

They were after her.

I assume “after” here means “in pursuit of,” and does not refer to temporal location, as “they were the people who filled the position of secretary after her.”
I hope the former is what “hin” is doing. I have been struggling with how adverbs nuance the meaning of verbs in sentences. Adverbs seem to make the difference between sentences ‘made up’ to learn a language, and real, complicated language.

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You are right, but it is the "her " that makes the difference:

“Sie waren hinter ihr her”.
They were chasing her.
(this might be boys in love, policemen in pursuit of a bank robber or headhunters for an excellent CIO)
“Sie waren hinter ihr.”
They were behind her.
(e.g. in a queue)
Neither sentence would be used temporally.

“Sie waren nach ihr im Ziel.”
They reached the finishing line after her.
I cannot explain why, but you cannot say “Sie waren nach ihr” alone without anything following.

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English Translation

They were after her.

It occurs to me that “her” in the examples functions a bit like “going” expresses motion in English, i.e., “they were going after her.”