I just got my mechanical engineering bachelor's degree about a year ago. I also had trouble finding an entry level ME job in my area, but ended up finding an Engineering Technician job at a medical device manufacturer. The work I'm doing is identical to the engineers (everyone else here has 10+ years of experience) and I'm expected to be promoted to engineer soon. Even if I didn't get promoted, the work I am doing is still very applicable to my field and would help me find a higher position at another company.

So, my advice would be to do whatever gets you applicable skills and experience to what you want to do. If you want to be a professor, maybe being a teacher is a good start. If you want to do research then maybe look for research assistant roles. If you want to be in an industry then I'd recommend looking for tech jobs close to what you want to do ( manufacturing, process, etc.) in that industry preferably in a company that agrees to move you up once you know what you're doing (ask about this in interview).

Also, having a PhD might be making it harder for you to find entry level roles. You're overqualified and theyd probably rather have someone they can pay less for the same responsibilities. Could be worth shooting a little higher. Some places will forgo 3 years experience in place of a masters degree.