401k Forfeiture Contribution

I’m going to be leaving my current company and have a considerable amount of unvested funds in the company’s retirement plan – both employer matching contributions and non-elective employer contributions. Fortunately, the company I’m joining has agreed to “buy me out” of the unvested amounts I’d be forfeiting in the form of a sign-on bonus. The bonus will be taxable and paid as part of my first paycheck but that doesn’t do me a whole lot of good from the standpoint of replenishing my retirement savings. Is there any way I can get the buy-out cash from the new employer to the tax-deferred side of the ledger to replenish the funds that were forfeited? Perhaps some kind of excess contribution to an IRA after I roll the old 401k over? Ideally, I’d like to have the same assets on the tax deferred retirement side of the ledger as I did before I left my soon-to-be former company but I’m not sure if this is possible.

Thanks in advance!