I think it is most likely I get into schools around the level of UIUC and UConn. Do you think these schools are still high enough in the rankings for my personal case?
I don't know enough about your personal case to make a determination. It's impossible to say "If you go to School X you will work in international law, and if you go to School Y you will not." I'm sure that there are graduates from just about every law school who work in international law, so I'm speaking in general terms only.
That said, there are some things to keep in mind, and which are applicable to any applicant.
1) Big Firms
Much of the international law market involves big firms with offices in LA, NYC, London, Shanghai, etc. These jobs are highly, highly competitive, and if you look at the firm profiles you will see that many of the lawyers are graduates of elite law schools. They like to hire graduates of internationally recognized schools like Harvard and Stanford because it impresses their international clients.
2) International Organizations/NGOs
Pretty much the same story. Highly competitive, lots of applicants, and a preference for elite pedigrees.
Now, does this mean that you can't practice international law unless you graduate from an elite school? Of course not! I know a woman who graduated from a lower tiered school here in CA and who is practicing internationally helping a foreign government to organize their own legal system. However, I think her case is exceptional. My point is simply that in any highly competitive market where the employer has the luxury of choosing from among many highly qualified applicants, pedigree can matter. As far as I can tell, this seems to be truer for some jobs than for others, and international law is one of those jobs.
I would contact the law schools you are interested in attending and ask them directly about international employment, internship opportunities, etc. Ask about alumni working internationally. Also research and contact firms and organizations that you are interested in, ask where they hire from and what they look for in applicants. Don't rely on anything you read here or elsewhere from me or anyone else, it's just our personal opinions. Get the information straight from the source. Your European LL.B is probably considered a very positive asset, and may give you a better chance of getting into international law than the average student.