“EM Brand Hurts Job Seekers”
Depends on what you did there and who’s hiring you.
There are plenty of places (ex. right-wing think tanks) that are happy to hire people from places like Exxon who have PR and/or legal backgrounds. Apparently, being the CEO of Exxon somehow qualifies you to lead the US State Department, so may there’s some opportunities there as well.
On the technical side, there isn’t much of a market for the skill set beyond the other super-majors. That’s why technical people who leave end up doing the same job at a similar company for a slight pay bump.
Where the OP is correct is with junior/middle managers. They’re widely seen as enforcers of a toxic status quo, and with companies struggling to retain workers, Exxon “leadership experience” is the last thing anyone needs in their company right now.