diff --git a/.github/workflows/smoke.yaml b/.github/workflows/smoke.yaml
index cb27552aa8..d5e5e64363 100644
--- a/.github/workflows/smoke.yaml
+++ b/.github/workflows/smoke.yaml
@@ -25,6 +25,7 @@ jobs:
       SHA: ${{ github.sha }}
     # TODO: drop PR testing under Windows because it's too slow?
     # if: github.event_name != 'pull_request' && matrix.os != 'windows-latest'
+    if: github.event_name != 'push' || github.repository == 'google/docsy'
     steps:
       - uses: actions/checkout@v4
       - uses: actions/setup-node@v4
diff --git a/tools/make-site.sh b/tools/make-site.sh
index 29c0415999..67c9435d27 100755
--- a/tools/make-site.sh
+++ b/tools/make-site.sh
@@ -142,13 +142,22 @@ function _set_up_site_using_hugo_modules() {
   else
     echo "[INFO] Fetch Docsy GitHub repo '$DOCSY_REPO' @ '$DOCSY_VERS'"
     mkdir tmp
-    BRANCH_SPEC=""
+    local BRANCH_SPEC=""
+    local DEPTH=10
+    local SWITCH_NEEDED=
+    local CLONE="git clone --depth=$DEPTH https://github.com/$DOCSY_REPO tmp/docsy"
     if [[ -n $DOCSY_VERS ]]; then
       BRANCH_SPEC="-b $DOCSY_VERS"
     fi
-    git clone --depth=1 https://github.com/$DOCSY_REPO $BRANCH_SPEC tmp/docsy
-    (cd tmp/docsy && git log -1)
-
+    if ! $CLONE $BRANCH_SPEC; then
+      SWITCH_NEEDED=1
+      $CLONE
+    fi
+    ( \
+      cd tmp/docsy && \
+      git log --oneline -$DEPTH && \
+      if [[ -n $SWITCH_NEEDED ]]; then git switch --detach $DOCSY_VERS; fi \
+    )
     echo "replace github.com/$DOCSY_REPO_DEFAULT => ./tmp/docsy" >> go.mod
     eval "$HUGO mod get github.com/$DOCSY_REPO_DEFAULT" $OUTPUT_REDIRECT
   fi