What can I do now, as a high school senior, to help me get more involved in the engineering world?

While in my Mechanical Engineering program, my roommate was in the Mathematics program. While we had in common courses including Calculus and Differential equations, he took much more advance courses regarding statistics, number theory, etc. He didn't have any of the core engineering courses that I had from year 2 through graduation. While math is fundamental to every engineering course, engineering practices are not core to a Math program. You may have a difficult time entering a Masters level engineering degree with none of the fundamentals. With your university not offering an engineering degree, you may not have an option to take engineering courses as electives.

There are interesting opportunities for Math majors related to areas like insurance and investments risk management. If you are targeting into a computer science type masters, that may fit well if you are able to take programing electives.

For your question, you should talk to your universities job placement office. Ask them where the math majors place after graduation. Which universities graduate programs, what companies, and what kinds of jobs. That will give you a good feel if you'll enjoy the outcome of their 4-5 year program. What you find will strongly influence the options you have when graduating yourself, and frame the difficulty of your potential engineering path. Don't be afraid of changing colleges if the one you are in is not going to get you where you want to be. It is far easier to transfer after Freshman year when you have fewer technical specific courses. After year 2, you are at a lot more risk of classes not counting towards your major and postponing graduation.

Even after all this angst over degrees and programs, never forget your career is what you make of it. Many people angle into different roles in their career that wouldn't seem a natural fit. I've seen people with philosophy degrees leading engineering teams. Getting the right degree just shortens the path and prepares you better. Consistently stay focused on long term goals that you work towards every day.

While in Highschool, you should consider joining community teams that work on technical projects. Look for a local robotics club, join a programing forum developing and sharing code, and take advantage of your local community college. Community colleges will often have clubs you can join of a technical nature. The net is actively participate in engineering type activity which will develop some of your skillset. It won't likely teach you the science and supporting math, but will give you practical hands on experience.

I am not sure about high school involvement. But after your freshman year in college, look for summer internships with consulting firms and city public works departments. This is an excellant way to gain exposure to the engineering field.