diff --git a/arq/connections.py b/arq/connections.py
index e843ac47..c1058890 100644
--- a/arq/connections.py
+++ b/arq/connections.py
@@ -167,9 +167,7 @@ async def enqueue_job(
             else:
                 score = enqueue_time_ms
 
-            expires_ms = expires_ms or (
-                1 if score - enqueue_time_ms < 1 else score - enqueue_time_ms + self.expires_extra_ms
-            )
+            expires_ms = expires_ms or score - enqueue_time_ms + self.expires_extra_ms
 
             job = serialize_job(function, args, kwargs, _job_try, enqueue_time_ms, serializer=self.job_serializer)
             pipe.multi()
diff --git a/arq/worker.py b/arq/worker.py
index 4c33b677..545ccf1c 100644
--- a/arq/worker.py
+++ b/arq/worker.py
@@ -751,7 +751,8 @@ async def run_cron(self, n: datetime, delay: float, num_windows: int = 2) -> Non
                     job_id = f'{cron_job.name}:{to_unix_ms(cron_job.next_run)}' if cron_job.unique else None
                 job_futures.add(
                     self.pool.enqueue_job(
-                        cron_job.name, _job_id=job_id, _queue_name=self.queue_name, _defer_until=cron_job.next_run
+                        cron_job.name, _job_id=job_id, _queue_name=self.queue_name,
+                        _defer_until=(cron_job.next_run if cron_job.next_run > datetime.now(tz=self.timezone) else None),
                     )
                 )
                 cron_job.calculate_next(cron_job.next_run)