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The answer is no – you don’t have to volunteer to repay it, anyway. A tribunal award is basically a guess about what the future will hold. You must tell the truth at the hearing about your prospects. But if you win compensation on the assumption that you’ll be out of work for another 6 months, and then the week after the hearing you land a new job that’s better paid than the old, that’s just your good luck.
But don’t crow about it to your old employer. It’s just conceivable that they might apply to the tribunal to review the remedy decision on the basis that there’s new evidence available that should change the award. They’d probably fail, but best not to risk it.