Good advice so far. I've been in the aerospace/defense industry (structural design and flight test) for about 10 yrs. I have a BS and MS in Mech E, and have my PPL. While the big companies are hiring (LM, Boeing, etc.), be weary because many can lay off just as quick as they hire. I too have worked in both gov't (NASA) and private flight test sectors. I've recently left the military side of the house and gotten into civil aviation...it has been MUCH more lucrative for me due to our customer base.
Degrees like EE and ME are broad, and don't pigeonhole you into a single field...so you are definitely on the right track!
As has been said, it's not hard to get into FTE...I have found that university/technical career fairs are a good way to network and get face time with company reps. However, I did get my current job solely off my LinkedIn page.
If you have your heart set on being a test pilot, why not look at the military (as someone mentioned)...you are still young enough for all branches (depending on how long you have left to complete your 4 yr degree)...and cant beat the training.
Good luck!