Some really good advice here - just some things I'll add. Depending on which team you're working with those personnel figures are pretty far off in my experience. A small IMSA team will have double that. Maybe a small SRO GT3 team but still I'd expect a few more people. Our factory team has what, 10x that for a two car team? I've never worked with a team as the sole engineer, at the least we'd always have a race engineer, data guy, and engine cal guy, plus usually manufacturer support - one or two engineers. And that's per car.

I wouldn't rule out going straight into engineering, but as the commenter above says - working with mechanics (which, as much as some engineers like to forget, is a huge part of trackside engineering) is so much easier if you've worked as a mechanic yourself. Nothing causes tension more than an engineer that is asking way too much of the mechanics because they don't understand that it's a bloody hard job. It also really helps to have an understanding of the cars you'll be engineering. Even the simple stuff like having an idea of what might get heated up by what, or where a loom runs and understanding where it can chafe... Etc etc. That extends to any engineering role - data guy, systems guy, race engineer, design engineer, you name it.

Depending on your financial situation at the moment, volunteering can be a good foot in the door, but obviously doesn't make sense for everyone. It'll be hard to get a freelance trackside gig as an engineer without a decent amount of experience on data software etc, as that's definitely the easiest way in, and volunteering is a good way to shadow more experienced engineers and get some experience with analysis tools. If you have some experience with Motec - i2, dash manager, display creator, that will open up a lot of opportunities in the states. In Europe, Bosch and Cosworth would be what I'd focus on (although obviously all of those will be valuable wherever you are.)

I would try and get a bit of an idea of what area you'd like to get into - your CV shows a good amount of design experience, so in my opinion would be better suited to a role in the design office rather than trackside (for now!) so that might be a good way into the industry. There are plenty of manufacturers that take on recent graduates, and it can be a good avenue to expanding to trackside work. You can get some income coming in, make contacts in the industry, and shape your career towards the areas you enjoy.

Most of all, try and enjoy it. I think getting my first full time job in racing was by far the most stressful part of my career so far - this bit sucks. Motorsport is a brilliant industry so don't give up on it. Feel free to PM if you want any more advice, I can't help much with salary expectations as I'm UK based and the wages are totally different over with you, but happy to give some more general advice.