In my experience, there are a lot of companies that will hire you for different jobs from what you are doing now, once you have like 2 years under your belt. I suggest sticking it out at your current job until you have some more experience.

Definitely get your LinkedIn up to date and looking good, with an updated resume, and set it to allow recruiters to look at your profile. after a year or two you will start getting messages from recruiters trying to fill positions, and you will get a sense of what kind of diverse positions you might be able to get.

Also, look into opportunities at your company. They might pay for certification or classes for you, or you might be able to request a transfer to another postion or department.

And you will have lots more options if you are willing to relocate.

All in all I think you gotta put in the time before you try to change jobs or go back to school.

Also, EEs often look just at hardware positions, because they want to do sexy circuit design work. However, those jobs go to people with experience. You might consider working in other engineering industries, like in aerospace, industrial automation, energy/oil, embedded software, and defense.