diff --git a/integration/integration_test.go b/integration/integration_test.go
index d69f4bd945b81..c26e08e49cd33 100644
--- a/integration/integration_test.go
+++ b/integration/integration_test.go
@@ -2403,7 +2403,7 @@ func testInvalidLogins(t *testing.T, suite *integrationTestSuite) {
 	require.NoError(t, err)
 
 	err = tc.SSH(context.Background(), cmd, false)
-	require.ErrorIs(t, err, trace.NotFound("failed to dial target host\n\tcluster \"wrong-site\" is not found"))
+	require.ErrorIs(t, err, trace.NotFound("failed to dial target host\n\tlooking up remote cluster \"wrong-site\"\n\t\tnot found"))
 }
 
 // TestTwoClustersTunnel creates two teleport clusters: "a" and "b" and creates a
diff --git a/lib/auth/auth_with_roles.go b/lib/auth/auth_with_roles.go
index a658dae4447cf..a6aba628ec48f 100644
--- a/lib/auth/auth_with_roles.go
+++ b/lib/auth/auth_with_roles.go
@@ -4526,7 +4526,7 @@ func (a *ServerWithRoles) GetRemoteCluster(clusterName string) (types.RemoteClus
 	}
 	cluster, err := a.authServer.GetRemoteCluster(clusterName)
 	if err != nil {
-		return nil, trace.Wrap(err)
+		return nil, utils.OpaqueAccessDenied(err)
 	}
 	if err := a.context.Checker.CheckAccessToRemoteCluster(cluster); err != nil {
 		return nil, utils.OpaqueAccessDenied(err)
diff --git a/lib/proxy/router.go b/lib/proxy/router.go
index c270012c650f5..98bef2451e02d 100644
--- a/lib/proxy/router.go
+++ b/lib/proxy/router.go
@@ -223,7 +223,7 @@ func (r *Router) DialHost(ctx context.Context, clientSrcAddr, clientDstAddr net.
 	if clusterName != r.clusterName {
 		remoteSite, err := r.getRemoteCluster(ctx, clusterName, accessChecker)
 		if err != nil {
-			return nil, targetTeleportVersion, trace.Wrap(err)
+			return nil, targetTeleportVersion, trace.Wrap(err, "looking up remote cluster %q", clusterName)
 		}
 		site = remoteSite
 	}
@@ -331,12 +331,12 @@ func (r *Router) getRemoteCluster(ctx context.Context, clusterName string, check
 
 	site, err := r.siteGetter.GetSite(clusterName)
 	if err != nil {
-		return nil, trace.Wrap(err)
+		return nil, utils.OpaqueAccessDenied(err)
 	}
 
 	rc, err := r.clusterGetter.GetRemoteCluster(clusterName)
 	if err != nil {
-		return nil, trace.Wrap(err)
+		return nil, utils.OpaqueAccessDenied(err)
 	}
 
 	if err := checker.CheckAccessToRemoteCluster(rc); err != nil {
diff --git a/lib/reversetunnelclient/api_with_roles.go b/lib/reversetunnelclient/api_with_roles.go
index 4098143fe8ab2..219c64f4535f0 100644
--- a/lib/reversetunnelclient/api_with_roles.go
+++ b/lib/reversetunnelclient/api_with_roles.go
@@ -86,14 +86,14 @@ func (t *TunnelWithRoles) GetSites() ([]RemoteSite, error) {
 func (t *TunnelWithRoles) GetSite(clusterName string) (RemoteSite, error) {
 	cluster, err := t.tunnel.GetSite(clusterName)
 	if err != nil {
-		return nil, trace.Wrap(err)
+		return nil, utils.OpaqueAccessDenied(err)
 	}
 	if t.localCluster == cluster.GetName() {
 		return cluster, nil
 	}
 	rc, err := t.access.GetRemoteCluster(clusterName)
 	if err != nil {
-		return nil, trace.Wrap(err)
+		return nil, utils.OpaqueAccessDenied(err)
 	}
 	if err := t.accessChecker.CheckAccessToRemoteCluster(rc); err != nil {
 		return nil, utils.OpaqueAccessDenied(err)
diff --git a/lib/utils/utils.go b/lib/utils/utils.go
index 000ad28d4ac37..7473e8e62badd 100644
--- a/lib/utils/utils.go
+++ b/lib/utils/utils.go
@@ -424,10 +424,11 @@ func IsCertExpiredError(err error) bool {
 	return strings.Contains(trace.Unwrap(err).Error(), "ssh: cert has expired")
 }
 
-// OpaqueAccessDenied returns a generic NotFound instead of AccessDenied
-// so as to avoid leaking the existence of secret resources.
+// OpaqueAccessDenied returns a generic [trace.NotFoundError] if [err] is a [trace.NotFoundError] or
+// a [trace.AccessDeniedError] so as to avoid leaking the existence of secret resources,
+// for other error types it returns the original error.
 func OpaqueAccessDenied(err error) error {
-	if trace.IsAccessDenied(err) {
+	if trace.IsNotFound(err) || trace.IsAccessDenied(err) {
 		return trace.NotFound("not found")
 	}
 	return trace.Wrap(err)