Any recommendations for quiet but reasonably close neighborhoods and/or specific buildings?
You might try this website from UTK: https://offcampushousing.utk.edu/.
A lot of student housing is in the Fort Sanders neighborhood, to the north of campus. Much of it is within walking distance of campus. While it can really vary from block to block, a lot of condos/apartments have been built there in the past 10 years. If you are going to be visiting campus, you might take a walk/drive through the neighborhood. Some rental units will be available by word of mouth, also.
Thanks!
@happymomof1: I am going to send you a PM, although it may be tomorrow when I get to it.