On Sundays through Fridays, run daily at 03:01 am.

On Saturdays, run daily and weekly sequentially, starting at 03:01 am.

This prevents daily and weekly from overlapping, while running weekly as
early as possible  (i.e. as soon as daily finishes) to give it time to
finish before monthly starts at 05:30 am.

It's probably possible to do something similar with monthly, making it
run as soon as daily (and possibly weekly, if the first of the month
is a Saturday) finishes, but this is left as an exercise for the reader.

etc/crontab

@@ -14,8 +14,8 @@ HOME=/var/log
 0	*	*	*	*	root	newsyslog
 #
 # do daily/weekly/monthly maintenance
-1	3	*	*	*	root	periodic daily
-15	4	*	*	6	root	periodic weekly
+1	3	*	*	0-5	root	periodic daily
+1	3	*	*	6	root	periodic daily weekly
 30	5	1	*	*	root	periodic monthly
 #
 # time zone change adjustment for wall cmos clock,