It's a bit hard to be absolutely sure what the appropriate word would be in this particular case, because it is not clear who are the parties, whether it is a standardised contract or a context-specific contract. However, that said, I would say that using 'each' or 'each respective' would be satisfactory, because 単位 just makes it clear in the contract that the particular day referred to will be different for each location/group/individual.
"...at each respective secondment location"
"...for each respective organisation/individual"

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Note added at 18 hrs (2018-01-12 00:52:56 GMT)
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possibly, you might consider using this:
'(on/from) the first day there (exists/ is any instance of) a clash with the restriction period applying to an individual client location'
(on/from) the first day (a clash occurs/there is any instance of a clash) with a restriction period applying to (a/any particular) group/person'
Its kind of hard to tell without know the whole context. Good luck.

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Note added at 18 hrs (2018-01-12 00:53:35 GMT)
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without *knowing*, that is.