diff --git a/lib/DateTimeUtils.test.ts b/lib/DateTimeUtils.test.ts
index b5f6149..cb84d8b 100644
--- a/lib/DateTimeUtils.test.ts
+++ b/lib/DateTimeUtils.test.ts
@@ -34,5 +34,24 @@ describe('DateTimeUtils.timestampToString', () => {
 
   it('still formats a valid time-of-day', () => {
     expect(DateTimeUtils.timestampToString(4980)).toBe('01:23:00');
+    expect(DateTimeUtils.timestampToString(86399)).toBe('23:59:59');
+  });
+
+  it('formats a full timestamp with its date', () => {
+    expect(DateTimeUtils.timestampToString(1704164645)).toBe(
+      '2024-01-02T03:04:05Z',
+    );
+  });
+});
+
+describe('DateTimeUtils.convertDayTimeToTod', () => {
+  it.each([
+    ['01000000', 'YYYY-MM-01T00:00:00'],
+    ['15123456', 'YYYY-MM-15T12:34:56'],
+    ['31235959', 'YYYY-MM-31T23:59:59'],
+  ])('keeps the day of month when formatting %s', (time, expected) => {
+    expect(
+      DateTimeUtils.timestampToString(DateTimeUtils.convertDayTimeToTod(time)),
+    ).toBe(expected);
   });
 });
diff --git a/lib/DateTimeUtils.ts b/lib/DateTimeUtils.ts
index 9912829..075ee55 100644
--- a/lib/DateTimeUtils.ts
+++ b/lib/DateTimeUtils.ts
@@ -86,9 +86,10 @@ export class DateTimeUtils {
       // only time, no date
       return date.toISOString().slice(11, 19);
     }
-    if (time < 2678400) {
-      // unknown month and date
-      return `YYYY-MM-${date.toISOString().slice(8, 19)}`;
+    if (time < 32 * 86400) {
+      // unknown month, day of month is time / 86400
+      const day = String(Math.floor(time / 86400)).padStart(2, '0');
+      return `YYYY-MM-${day}T${date.toISOString().slice(11, 19)}`;
     }
     //strip off millis
     return date.toISOString().slice(0, -5) + 'Z';