Let's say it's 6:00, and you hit the snooze button. Save the current value of the minute minus 1, into a comparitor, Trigger the alarm when the comparitor goes off automatically.
If you save the current value into the comparitor, the alarm triggers immediately. Now you need to add delay logic. Just use current minus 1, since it's only a few half-adders.
You can't use the high order minute digit because then your snooze length will depend on your current low order minute digit. Any logic to compensate starts adding a multiple of the number of components required.