Edit: Jakub Narebski has more git-fu. The following much simpler command works perfectly:
git describe --tags
(Or without the --tags if you have checked out an annotated tag. My tag is lightweight, so I need the --tags.)
original answer follows:
git describe --exact-match --tags $(git log -n1 --pretty='%h')
Someone with more git-fu may have a more elegant solution...
This leverages the fact that git-log reports the log starting from what you've checked out. %h prints the abbreviated hash. Then git describe --exact-match --tags finds the tag (lightweight or annotated) that exactly matches that commit.
The $() syntax above assumes you're using bash or similar.