If you want to extract just a simple piece of information, you can get that using git show with the --format=<string> option...and ask it not to give you the diff with --no-patch. This means you can get a printf-style output of whatever you want, which might often be a single field.
For instance, to get just the shortened hash (%h) you could say:
$ git show --format="%h" --no-patch
4b703eb
If you're looking to save that into an environment variable in bash (a likely thing for people to want to do) you can use the $() syntax:
$ GIT_COMMIT="$(git show --format="%h" --no-patch)"
$ echo $GIT_COMMIT
4b703eb
The full list of what you can do is in git show --help. But here's an abbreviated list of properties that might be useful:
%H commit hash%h abbreviated commit hash%T tree hash%t abbreviated tree hash%P parent hashes%p abbreviated parent hashes%an author name%ae author email%at author date, UNIX timestamp%aI author date, strict ISO 8601 format%cn committer name%ce committer email%ct committer date, UNIX timestamp%cI committer date, strict ISO 8601 format%s subject%f sanitized subject line, suitable for a filename%gD reflog selector, e.g., refs/stash@{1}%gd shortened reflog selector, e.g., stash@{1}