Okay, the circumstances are this if I understand it correctly. You were about to cut the employee's hours from full to part time. This was not his idea, was your idea, based on your company's decision in its own best interest. This may have been a little distressing to the employee. Fair? Not an issue. You the employer made the decision, gave the employee no choice except to accept this new part time job situation or quit and find another one. The employee voluntarily quits the job, probably says he needs a full-time job. He finds a full time job shortly after leaving your employment, has worked there until the present, when he is laid off due to no fault of his own from the new employer. This may be a temporary lay off for a few weeks, or it may be a permanent one, you probably were not given this information. Probably you were notified that you will be liable for a certain percentage of the claim, a maximum possible charge of......because there's no way to determine yet whether the person is going to draw out all the claim or just a few weeks of it. This may or may not cause your tax rate to be raised.
You receive notice from the charge unit of your state unemployment system that a sizeable portion of the former employee's claim will be charged against your account. There is probably some sort of information on this notice about how to submit an appeal, some way that you can inform the charging unit that the employee voluntarily quit your job, and therefore you request a non-charge to your employer tax account.
The problem is that the person has already explained the circumstances under which he left your job to the claims office when he filed the unemployment claim, and it was likely determined that he actually left your job for a "good, non-misconduct reason." That was that the hours were cut to part-time, which is actually the decision of the employer, not considered the fault of the employee. I'd say you're probably on the hook for these unemployment benefits, but it wouldn't hurt to ask the charge unit and see if you can appeal.
If the employee's hours were reduced significantly, he could probably have gotten approved for benefits at the time he left your employment. But the employee went on quickly and went right to work somewhere. Unemployment benefits isn't very much money if the person can find anything else, so perhaps he won't be drawing long. Hopefully the next employer won't keep the person laid off too long before a recall, or the person will move on to something else full time.