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Thank you for your reply. The law does not define "new hire" for this purpose. However, so long as the purpose of your drug test is because you were hired, and not simply "random" then it is going to be legal. Here, even though some time has passed, the purpose of this drug test is very clearly because you were hired as that is your understanding and also your employer's policy. There is no contesting the legality of a drug test on this basis.
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Very best wishes.