Went on foot while a saint got a lift (6)

I believe the answer is:

strode

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'went on foot' is the definition.
(I've seen this before)

'saint got a lift' is the wordplay.
'saint' becomes 'st' (abbreviation).
'got a lift' becomes 'rode' (I've seen this in another clue).
'st'+'rode'='STRODE'

'while a' acts as a link.
I am not very happy about this link. Some or all of it may be part of another bit of the clue.

Can you help me to learn more?

(Other definitions for strode that I've seen before include "Galumphed" , "Walked fast with long steps" , "Paced" , "went on foot" .)

I've seen this clue in the Evening Standard.

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