How much does having a good program at your school, in your chosen specialty, matter?

For instance, my school is relatively mid tier like t-50, but has a really great, high ranking, ENT program (like top 10 in the us). My specialty of choice is also ENT. How much does having a good program matter in terms of matching, especially when coming from a mid tier school. Is the program more important due to LORs, connections, etc, or does the name of the school take more precedent?