In any contract, you can describe certain sections which are specifically called out as continuing even if the rest of the contract is terminated. So the non-compete clause would be specified as something which remains in force for a certain period of time even if the employment contract is terminated.
Alternatively, I’ve sign confidentiality agreements and non-disclosure agreements with employers as separate agreements. I remained an at-will employee (e.g. without an employment contract) but was still bound by those agreements. I suspect you can also be asked to sign a non-compete agreement separately, although there are limitations on how long such agreements can run.
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