why not just append day and time to the filename then???
for example in your script:
bkup_filename="my_backup_`/bin/date +"%Y_%h_%d_%T"`
This will create a string akin to "my_backup_2009_May_18_21:43:23"
Look at the other options to the date command to tailor this as you will... There is also the %s option that will print the seconds since the start of the unix epoch ( Jan 1, 1970). That can serve as your single number.
--Eric