Supercap for Backup Power (10 hours Backup Time Min.)
I believe that your best bet is to use a dedicated RTC chip for the clock function. These can connect to your microcontroller via an I2C bus and thus take very few I.O pins to support. If fact if you have already deployed the the I2C bus for other purposes then supporting the RTC chip takes no additional pins.

The reason for using an RTC chip over trying to backup your microcontroller is the difference in achievable standby operating current. RTC chips are optimized for extremely low power operation, particularly if you deploy them with their own attached 32.768 KHz crystal. 

I have also found that the external RTC chips with crystal can achieve far better long term accuracy. In fact, if you select the right RTC chip you can use internal registers to trim in the oscillator to get even less clock drift over time.

The use of super capacitors is OK but in my experience, with a very low power RTC chip, that it may very well be cheaper and take less board space to use a small coin cell to power the RTC chip. A soldered in battery holder for a BR1225 coin cell will let you slide in the cell and take board space less than .5x.5". Such cell with 48mAH capacity will keep the RTC supported for years. Even better, with the proper RTC chip selection, is that the chip provides a separate pin for the battery connection so it provides all the regular VCC rail to battery switch over circuitry on die.