From my knowledge, times.conf is just for configuring time ranges. It sounds like you require a number of different TimeRangePickers. I believe that there is only one TimeRangePicker module at the moment so you will not be able to achieve what you are asking (although this could be very handy). Maybe this could be done by creating a custom module but this may be a lot of work.
For now, I would just set each screen to have a default value for the TimeRangePicker.