diff --git a/Library/Homebrew/cmd/update.sh b/Library/Homebrew/cmd/update.sh
index 579890820d..4d72c66112 100644
--- a/Library/Homebrew/cmd/update.sh
+++ b/Library/Homebrew/cmd/update.sh
@@ -217,7 +217,8 @@ merge_or_rebase() {
 
   if [[ "$DIR" = "$HOMEBREW_REPOSITORY" && -z "$HOMEBREW_NO_UPDATE_CLEANUP" ]]
   then
-    UPSTREAM_TAG="$(git tag --list --sort=-version:refname | grep '^[0-9]*\.[0-9]*\.[0-9]*$' | head -n1)"
+    UPSTREAM_TAG="$(git tag --list --sort=-version:refname |
+                    grep --max-count=1 '^[0-9]*\.[0-9]*\.[0-9]*$')"
   else
     UPSTREAM_TAG=""
   fi
@@ -244,7 +245,7 @@ merge_or_rebase() {
              stash save --include-untracked "${QUIET_ARGS[@]}"
     then
       odie <<EOS
-Could not `git stash` in $DIR!
+Could not 'git stash' in $DIR!
 Please stash/commit manually if you need to keep your changes or, if not, run:
   cd $DIR
   git reset --hard origin/master
@@ -469,15 +470,30 @@ EOS
       then
         UPSTREAM_REPOSITORY="${UPSTREAM_REPOSITORY_URL#https://github.com/}"
         UPSTREAM_REPOSITORY="${UPSTREAM_REPOSITORY%.git}"
-        UPSTREAM_BRANCH_LOCAL_SHA="$(git rev-parse "refs/remotes/origin/$UPSTREAM_BRANCH_DIR")"
-        # Only try to `git fetch` when the upstream branch is at a different SHA
-        # (so the API does not return 304: unmodified).
+
+        if [[ "$DIR" = "$HOMEBREW_REPOSITORY" && -z "$HOMEBREW_NO_UPDATE_CLEANUP" ]]
+        then
+          # Only try to `git fetch` when the upstream tags have changed
+          # (so the API does not return 304: unmodified).
+          GITHUB_API_ETAG="$(sed -n 's/^ETag: "\([a-f0-9]\{32\}\)".*/\1/p' ".git/GITHUB_HEADERS")"
+          GITHUB_API_ACCEPT="application/vnd.github.v3+json"
+          GITHUB_API_ENDPOINT="tags"
+        else
+          # Only try to `git fetch` when the upstream branch is at a different SHA
+          # (so the API does not return 304: unmodified).
+          GITHUB_API_ETAG="$(git rev-parse "refs/remotes/origin/$UPSTREAM_BRANCH_DIR")"
+          GITHUB_API_ACCEPT="application/vnd.github.v3.sha"
+          GITHUB_API_ENDPOINT="commits/$UPSTREAM_BRANCH_DIR"
+        fi
+
         UPSTREAM_SHA_HTTP_CODE="$("$HOMEBREW_CURL" --silent --max-time 3 \
            --output /dev/null --write-out "%{http_code}" \
+           --dump-header "$DIR/.git/GITHUB_HEADERS" \
            --user-agent "$HOMEBREW_USER_AGENT_CURL" \
-           --header "Accept: application/vnd.github.v3.sha" \
-           --header "If-None-Match: \"$UPSTREAM_BRANCH_LOCAL_SHA\"" \
-           "https://api.github.com/repos/$UPSTREAM_REPOSITORY/commits/$UPSTREAM_BRANCH_DIR")"
+           --header "Accept: $GITHUB_API_ACCEPT" \
+           --header "If-None-Match: \"$GITHUB_API_ETAG\"" \
+           "https://api.github.com/repos/$UPSTREAM_REPOSITORY/$GITHUB_API_ENDPOINT")"
+
         # Touch FETCH_HEAD to confirm we've checked for an update.
         [[ -f "$DIR/.git/FETCH_HEAD" ]] && touch "$DIR/.git/FETCH_HEAD"
         [[ -z "$HOMEBREW_UPDATE_FORCE" ]] && [[ "$UPSTREAM_SHA_HTTP_CODE" = "304" ]] && exit