The most common and most likely answer to this crossword clue is the 11 letter word HITCHHIKING
I've seen this crossword clue in The Times
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Buddy explains!
The straight part
We know that "seeking a lift" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case seeking a lift = hitchhiking.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does HITCHHIKING mean?
HITCHHIKING verb- To try to get a ride in a passing vehicle while standing at the side of a road, generally by either sticking out one's finger or thumb or holding a sign with one's stated destination.
- To be carried along with something else, for example Genetic Hitchhiking where a gene is propagated because it occurs in conjunction with a favourable mutation, or Cultural Hitchhiking where a cultural trait spreads with a technologically advanced population.