Lucky-Win-2943's journey

I was recently laid off from my position after nearly 6 years with the company due to budget cuts. I received about 2 weeks' notice and was offered 4 weeks of severance pay, contingent on signing a separation agreement and waiver of claims. I attempted to negotiate for a higher severance, specifically asking for 12 weeks based on my years of service, but HR informed me that their offer was already an exception to their standard policy. Now, I'm at a crossroads and wondering if I should negotiate again for a middle ground, like 8 weeks, or if I should just accept the 4 weeks and move on. I'm curious if anyone has had success in similar situations or if it's worth the effort to ask again.