diff --git a/src/init.c b/src/init.c
index ac0b1535..15c46478 100644
--- a/src/init.c
+++ b/src/init.c
@@ -795,10 +795,15 @@ void InitRemind(int argc, char const *argv[])
 
     }
 
-/* Figure out the offset from UTC */
+    /* Figure out the offset from UTC */
     if (CalculateUTC)
         (void) CalcMinsFromUTC(DSEToday, MinutesPastMidnight(0),
                                &MinsFromUTC, NULL);
+    /* Warn if REMIND_RUNNING_TEST is set */
+    s = getenv("REMIND_RUNNING_TEST");
+    if (s && !strcmp(s, "1")) {
+        fprintf(stderr, "WARNING: The REMIND_RUNNING_TEST environment variable is set to 1.\nThis is intended only for the Remind acceptance test and not normal use.\n");
+    }
 }
 
 /***************************************************************/
diff --git a/src/main.c b/src/main.c
index 07aa7774..45f986f7 100644
--- a/src/main.c
+++ b/src/main.c
@@ -1007,13 +1007,27 @@ int SystemTime(int realtime)
 {
     time_t now;
     struct tm *t;
+    int r;
+    char const *s;
 
     if (!realtime && (SysTime != -1)) return SysTime;
 
     now = time(NULL);
     t = localtime(&now);
-    return t->tm_hour * 3600L + t->tm_min * 60L +
+    r =  t->tm_hour * 3600L + t->tm_min * 60L +
         t->tm_sec;
+
+    if (r < 82800) {
+        /* Before 23:00 */
+        return r;
+    }
+
+    s = getenv("REMIND_RUNNING_TEST");
+    if (!s || strcmp(s, "1")) return r;
+
+    /* If it's after 23:00, subtract an hour to avoid
+       date rollover in queueing tests */
+    return r-3600;
 }
 
 /***************************************************************/
diff --git a/tests/test-rem b/tests/test-rem
index 26b7d521..17ff8353 100644
--- a/tests/test-rem
+++ b/tests/test-rem
@@ -37,13 +37,6 @@ alias remind="echo You should be using ../src/remind explicitly in test-rem >&2;
 TZ=UTC
 export TZ
 
-DO_QUEUE_TESTS=1
-RESULT=`(echo 'BANNER %'; echo 'IF now() > 23:55'; echo 'MSG late%'; echo 'ENDIF') | ../src/remind -h -`
-
-if test "$RESULT" = "late" ; then
-    DO_QUEUE_TESTS=0
-fi
-
 # If we're already in a utf-8 locale, do
 # nothing; otherwise, set LC_ALL
 OK=0
@@ -476,20 +469,11 @@ rm -rf include_dir/ww
 # Test queueing.  Because eventstart depends on the actual system
 # date, we have to convert it to some constant (in this case,
 # VOLATILE) so that tests are not dependent on the system date.
-if test "$DO_QUEUE_TESTS" = 1 ; then
-    echo JSONQUEUE | ../src/remind -z0 ../tests/queue1.rem 2>&1 | sed -e 's/"eventstart":"................"/"eventstart":"VOLATILE"/g' | sed -e 's/"qid":"[0-9a-f]*",//g' >> ../tests/test.out 2>&1
-    echo QUEUE | ../src/remind -zj ../tests/queue1.rem 2>&1 | sed -e 's/"eventstart":"................"/"eventstart":"VOLATILE"/g' | sed -e 's/"qid":"[0-9a-f]*",//g' >> ../tests/test.out 2>&1
-else
-    echo "*** Skipping queueing tests between 23:55 and 00:00 UTC"
-    echo "*** If you want to run these tests, wait until 00:00 UTC"
-    # Fake the output we expect from above
-    cat >> ../tests/test.out <<'EOF'
-NOTE JSONQUEUE
-[{"priority":2,"eventstart":"VOLATILE","time":"23:59","nexttime":"23:59","tdelta":0,"trep":0,"rundisabled":0,"ntrig":1,"filename":"../tests/queue2.rem","lineno":1,"type":"MSG_TYPE","body":"XXXX"},{"priority":999,"eventstart":"VOLATILE","time":"23:58","nexttime":"23:58","tdelta":0,"trep":0,"rundisabled":0,"ntrig":1,"filename":"../tests/queue1.rem","lineno":5,"type":"MSG_TYPE","body":"quux"},{"priority":42,"eventstart":"VOLATILE","time":"23:57","nexttime":"23:57","tdelta":0,"trep":0,"rundisabled":0,"ntrig":1,"filename":"../tests/queue1.rem","lineno":4,"type":"MSG_TYPE","body":"bar"},{"priority":5000,"eventstart":"VOLATILE","time":"23:56","nexttime":"23:56","tdelta":0,"trep":0,"rundisabled":0,"ntrig":1,"filename":"../tests/queue1.rem","lineno":3,"type":"MSG_TYPE","body":"foo"}]
-NOTE ENDJSONQUEUE
-{"response":"queue","queue":[{"priority":2,"eventstart":"VOLATILE","time":"23:59","nexttime":"23:59","tdelta":0,"trep":0,"rundisabled":0,"ntrig":1,"filename":"../tests/queue2.rem","lineno":1,"type":"MSG_TYPE","body":"XXXX"},{"priority":999,"eventstart":"VOLATILE","time":"23:58","nexttime":"23:58","tdelta":0,"trep":0,"rundisabled":0,"ntrig":1,"filename":"../tests/queue1.rem","lineno":5,"type":"MSG_TYPE","body":"quux"},{"priority":42,"eventstart":"VOLATILE","time":"23:57","nexttime":"23:57","tdelta":0,"trep":0,"rundisabled":0,"ntrig":1,"filename":"../tests/queue1.rem","lineno":4,"type":"MSG_TYPE","body":"bar"},{"priority":5000,"eventstart":"VOLATILE","time":"23:56","nexttime":"23:56","tdelta":0,"trep":0,"rundisabled":0,"ntrig":1,"filename":"../tests/queue1.rem","lineno":3,"type":"MSG_TYPE","body":"foo"}],"command":"QUEUE"}
-EOF
-fi
+REMIND_RUNNING_TEST=1
+export REMIND_RUNNING_TEST
+echo JSONQUEUE | ../src/remind -z0 ../tests/queue1.rem 2>&1 | sed -e 's/"eventstart":"................"/"eventstart":"VOLATILE"/g' | sed -e 's/"qid":"[0-9a-f]*",//g' >> ../tests/test.out 2>&1
+echo QUEUE | ../src/remind -zj ../tests/queue1.rem 2>&1 | sed -e 's/"eventstart":"................"/"eventstart":"VOLATILE"/g' | sed -e 's/"qid":"[0-9a-f]*",//g' >> ../tests/test.out 2>&1
+unset REMIND_RUNNING_TEST
 
 # Test for leap year bug that was fixed
 ../src/remind -dte - 28 Feb 2024 <<'EOF' >> ../tests/test.out 2>&1
diff --git a/tests/test.cmp b/tests/test.cmp
index 9c9cccfa..074fde80 100644
--- a/tests/test.cmp
+++ b/tests/test.cmp
@@ -23201,9 +23201,13 @@ Error reading include_dir/ww: Can't open file
 SECURITY: Won't read world-writable file or directory!
 Error reading include_dir/ww: No files matching *.rem
 05.02.03
+WARNING: The REMIND_RUNNING_TEST environment variable is set to 1.
+This is intended only for the Remind acceptance test and not normal use.
 NOTE JSONQUEUE
 [{"priority":2,"eventstart":"VOLATILE","time":"23:59","nexttime":"23:59","tdelta":0,"trep":0,"rundisabled":0,"ntrig":1,"filename":"../tests/queue2.rem","lineno":1,"type":"MSG_TYPE","body":"XXXX"},{"priority":999,"eventstart":"VOLATILE","time":"23:58","nexttime":"23:58","tdelta":0,"trep":0,"rundisabled":0,"ntrig":1,"filename":"../tests/queue1.rem","lineno":5,"type":"MSG_TYPE","body":"quux"},{"priority":42,"eventstart":"VOLATILE","time":"23:57","nexttime":"23:57","tdelta":0,"trep":0,"rundisabled":0,"ntrig":1,"filename":"../tests/queue1.rem","lineno":4,"type":"MSG_TYPE","body":"bar"},{"priority":5000,"eventstart":"VOLATILE","time":"23:56","nexttime":"23:56","tdelta":0,"trep":0,"rundisabled":0,"ntrig":1,"filename":"../tests/queue1.rem","lineno":3,"type":"MSG_TYPE","body":"foo"}]
 NOTE ENDJSONQUEUE
+WARNING: The REMIND_RUNNING_TEST environment variable is set to 1.
+This is intended only for the Remind acceptance test and not normal use.
 {"response":"queue","queue":[{"priority":2,"eventstart":"VOLATILE","time":"23:59","nexttime":"23:59","tdelta":0,"trep":0,"rundisabled":0,"ntrig":1,"filename":"../tests/queue2.rem","lineno":1,"type":"MSG_TYPE","body":"XXXX"},{"priority":999,"eventstart":"VOLATILE","time":"23:58","nexttime":"23:58","tdelta":0,"trep":0,"rundisabled":0,"ntrig":1,"filename":"../tests/queue1.rem","lineno":5,"type":"MSG_TYPE","body":"quux"},{"priority":42,"eventstart":"VOLATILE","time":"23:57","nexttime":"23:57","tdelta":0,"trep":0,"rundisabled":0,"ntrig":1,"filename":"../tests/queue1.rem","lineno":4,"type":"MSG_TYPE","body":"bar"},{"priority":5000,"eventstart":"VOLATILE","time":"23:56","nexttime":"23:56","tdelta":0,"trep":0,"rundisabled":0,"ntrig":1,"filename":"../tests/queue1.rem","lineno":3,"type":"MSG_TYPE","body":"foo"}],"command":"QUEUE"}
 BANNER %
 REM 29 MSG One