First of all, just because you attend a professional psychology school does not mean that you will not receive a top tier job. I attended a recently unaccredited school (Midwestern University-- we are now APA accredited which helped a lot come internship) and, due to excellent training and a decent personality while interviewing, I was fortunate enough to receive an APA internship at one of the top medical schools in the country. I will be working with nearly all Ph.D.s (all of whom seem lovely, I might add) and I have received job offers from my previous two training sites already. Moral of the story? The name of your school or the type of degree does not equal automatic mediocrity. I went out of my way to pursue excellent training at practicum sites and leadership positions at my school. Also, I have ZERO research experience (with the exception of collecting data-- no publications) and I still received 9 APA interviews for internship. I'm not posting this to brag, but to instead try to dispell this unfortunate notion that what type of degree and/or your school's name is the deciding factor in internships. Honestly, in my experience, internships cared a lot more about your passion, training experience, and, quite frankly, if you had a decent, non-abrasive personality that was pleasant and not pompous. Don't lose faith-- pursue your passion in this field to the utmost, become friendly and professional with your supervisors, and beef up on your interviewing skills.
On a separate note, also check out Roosevelt University in Chicago-- I know several people from that school and it seems like they are pretty good at prepping their students for internship. I would also recommend my school, Midwestern University, as they have made a great deal of changes and the training director goes out of her way to help you get exactly what you want. However, since this is our first year as an APA-accredited school, our APA-internship match rate is still not up to others' standards (however, we do have a 100% match rate to APPIC sites); like I previously stated, that hasn't stood in my way so don't let anyone tell you that you will be a mediocre psychologist just because you attend a professional school. There are plenty of poor clinicians who come from these "top notch" schools and fabulous clinicians from the schools that others scoff at. You just have to work extra hard to ensure that your training is superb regardless of your school.
Best of luck to you! Also, sorry for the long post!