prevent multiple tail in back ground

Dears

i have a scrip run in unix that need to use the tail -f command, as below:

DATE=`date '+%m%d%y'`
tail -f /var/messages |  grep "start" >> /export/logs/start_${DATE}.out

but the problem that the tail -f will be stop and working in the background in then end of the day (23:59:59) that mean the time (start_${DATE}.out) is finished. and the time that the messages files are rotated (2:59:59)... so in the crontab i make it it run twice, one at 00:00 and second is 3:30 . at that case i will have many old tail -f processes run in the background and i need to kill them manually... so is there any way to make the tail -f keep running even a new file is created with new date and even the file messages are rotated?

thanks alot for your support really appreciated