Is there a quick way to get the very last file in a large TAR?

No, unfortunately there is not. From Wikipedia

Another weakness of the tar format compared to other archive formats is that there is no centralized location for the information about the contents of the file (a "table of contents" of sorts). So to list the names of the files that are in the archive, one must read through the entire archive and look for places where files start. Also, to extract one small file from the archive, instead of being able to lookup the offset in a table and go directly to that location, like other archive formats, with tar, one has to read through the entire archive, looking for the place where the desired file starts. For large tar archives, this causes a big performance penalty, making tar archives unsuitable for situations that often require random access of individual files.