You can transfer up, but only if two things happen: 1.) you do very well during your first year at whatever college you matriculated to (this implies law review, top 15%, in many cases top 10% or 5%, depending on the school you're transferring to) and 2.) that your stats (undergrad GPA, lsat score) are strong enough to have made you competitive as a first-year applicant. You'll have some leway if you have a relatively low lsat score, but not enough to get you into harvard if your lsat was sub-165 (sub-170 really). Just do well wherever you end up at, and aim at schools that are ranked above it within 10-20 positions (it is a little difficult to jump tiers, but certainly possible). I can't think of any schools that don't average lsat scores, but I think Penn is one of them assuming you score at least 5 points above your previous score (in your case, you'll need to improve your lsat score by at least 30 points to be competitive for Penn and comparable schools. If those schools average your score, you'll have no chance at them unless they feel very generous). Good luck with your second lsat exam and your second round of law school apps. It sucks you have to go through this again. Hopefully this time will work out for you better.
Good advice for the most part, except one thing -- actually, to transfer up you don't have to have an LSAT score on par with admitted students (there are exceptions...but Harvard is the only one I know of). It makes sense if you think about it. The LSAT predicts your law school performance, and once you've earned grades in law school, the LSAT is kind of moot.
Otherwise, solid advice. I agree that in many cases you'll probably need better than top 15% to transfer, though I've heard some schools (Wash U comes to mind) will take you as long as you're at least top 25%.
Oh, and other schools that don't average your LSAT scores include Minnesota, Cornell and Pitt (there are probably many more I can't think of off the top of my head).