I’d suggest researching firms that you’d like to work for and emailing them directly to introduce yourself and your skillset. Sometimes, you end up contacting someone around the that they’re bidding for a project or have identified a need to hire a new person. If you fit the bill… it is much easier to just hire you than post a job, sort through resumes, and interview a whole bunch of other people!
For smaller firms, this is easy and you can be relatively sure that someone will at least read your email that you send to “info@…” (of course there are no guarantees on getting a response!). For larger firms, sending a general inquiry email to “careers@…” will probably just go straight to a trash folder. So, you might have to do some LinkedIn research to seek out and connect with manager/director level people in the department you’d want to be in. Include an enthusiastic (and not too boilerplate) message about your desire to work for company and their work, your resume and a pdf of your portfolio (or maybe just a shortened teaser version).
As a youngish ME with an ID grad degree myself, I have found it to be a relatively difficult spot to be in. When I was looking for a job in a new city a few years ago, I found that a lot of firms were looking for highly experienced product design ME talent, which I wasn’t. And if there were openings for Junior ID positions, my portfolio wasn’t competitive for those positions either. So, I just started cold contacting every firm in the area that I found interesting. And that is how I found my current job for an ME design firm. I do more ME than ID work at this time, but I’m slowly working towards ID. Gotta start somewhere!
So… I understand your situation. Good luck with the search!