Pretty likely, especially if your high PSG for your mid-career and if your function/BU is getting slashed and burned. More then likely you already had it too cozy at Chevron if you have been here long enough to collect a 1-year severance which means you have not been scanning the market for a while to see what life outside the Chevron bubble is like and what skills are marketable. You might not like what you find to be honest given that Chevron is one of the few places where you can use the minimal amount of brain power, and still be well compensated for your ignorance given the level of stupidity that runs rampant in the company. Just look at some of the VPs in UC if you doubt me.
I am guessing you are in some support function like Finance, HR, Legal, or IT if your field is hiring these days as those would be industry neutral. Best advice I can give you is to start applying now and go on interviews and see what the market is looking for these days and how you stack up comparatively, or at least get harsh reality of what you need to brush up on to be market competitive these days especially when your interviewers ask you questions and you start blanking on the basics your expected to know in your field and at your level of experience in the current market without muttering Chevron jargon.
My advice is regardless of how your first experience in the new world out there goes, continue to apply for jobs and once you get an offer for something you want, hopefully EOI at the same time. Best of luck whichever you decide bro.
What field are you in if I might ask?