If you just want the phone to charge in the sun why would you want to use supercaps or a battery? A super cap won't do much of anything useful as they can't by themselves maintain a voltage, if the charger in your phone is set to turn off at 4.0 volts the bulk majority of the charge in a super cap will never be used because between 4.0 volts and 0 volts there is a huge amount of energy stored that the phone can never tap, it will at best buffer clouds passing or passing under a tree, and most of it's potential will never be used, a battery pack could be used but again it's not really a plug and play thing, not used optimally at least, you'd be better off buying a commercial device that has all the circuitry required to properly do this.
I do question the need for this device in the first place, how long are you going to be out? I have an identical battery in my cell phone and I only have to charge it once every 3 days or so, and that's with moderate texting and minimal voice. If this is for safety there is a simpler sollution; what really drains a cell phone's batteries is when you're in a remote location with little signal, it will kill the battery trying to hunt down a tower connection, if the phone is off the battery will be preserved for weeks. Only turn the phone on when you need it, to check for voice mails or what not once per day if you can get a signal. If you're going to be out for more than a couple weeks bring a second battery fully charged and vacuum packed in a plastic bag. A cell phone battery will maintain a useful charge for months not connected to anything.
I've had problems when visiting my Grandmother because she lives in a mountainous region of PA, and her house happens to be an RF dead zone. Any cell phone brought to her house will have a dead battery in less than two hours from trying to find a tower, which it can occasionally connect to JUST long enough for it to want to keep trying.