You can do two things.
You can run the base of the joule thief transistor from a separate power source, like a fresh 1.5V battery, or a CR2032. Also, using a 10 ohm resistor plus a 1K (or 10K) potentiometer in place of the 1K resistor would help set the brightness. Because the battery is only powering the 1K resistor and the base of the transistor, it should last a long time.
You can bring back the voltage being produced right above the LED, and use that to power the transistor. For as long as the joule thief operates, the 3 volts from the LED will be produced, and allow the joule thief to operate well below 0.55 volts, especially with a supercapacitor, which typically has low internal resistance and therefore can operate better at lower voltages than a typical battery. If you combine this idea with the previous idea (and a diode), you can reduce battery usage.
I will try to add schematics to show what I'm talking about.
EDIT #2:
Here's a schematic for item 1 above (Click on it to see it larger):
Above you should see two circuits that are almost exactly the same, except for the modifications that I want to talk about, and resistances tuned so that the LED's start out at the same output so that we have a good comparison of the two circuits.
The circuit on the right powering LED D4 is the closest to your circuit, a standard Joule Thief.
The circuit on the left has modifications to allow the voltage of the LED D1 to be brought back into the transistor. The diodes just guarantee that current will flow properly (won't flow backwards).
The resistors have been tweaked to give a good comparison between the two circuits.
For the D1 circuit... The 1K resistor on the left was increased to make it deliver less of the base current to the transistor, so that its job is just to get the circuit started, and also get the LED D1 lit so that the approximately 3 volts is reached.
The horizontal red line is 0.5 Volts. The Aqua line is V(V1), which is exactly the same as V(V2), and is falling to show us how both Joule Thief circuits behave as the power source voltage falls. The place where the Aqua line intersects the red line is at the point in time when 0.5 Volts "happens", and I've used the cursors to get a reading off of both D1 and D4, to compare the currents through both LED's at a half volt in (pretending that the supercap is half a volt for both circuits). You should be able to see in the dialog box that for D4, the current is about 1.64mA, whereas the current in D1 is much better at around 5.04mA. You should notice the Green curve representing D1 is giving much better current for a given voltage, all other things being equal.
So, this mod to the standard Joule Thief circuit will use significantly more of the power that's stored up in your 1 Farad supercapacitor, allowing the LED to stay lit for a longer period of time.
Please realize that because the LED will stay brighter as voltage falls, it may actually use up the power in the supercap faster. So you may have to increase the Joule Thief resistance to start it out dimmer to begin with -- and it should maintain a better brightness as the supercap voltage falls.
Note that the mod to the Joule Thief only works once the circuit has started up. If the supercapacitor falls below 0.5 volts and you try to turn it back on, it probably won't work because there is no longer any voltage at LED D1 (it's off).
Through significant experimentation, I have found 2.6mA to deliver sufficient brightness for most tasks (in a dark area, with efficient LED's). It's surprising how much longer an LED will stay lit when run at the minimum necessary to do the job. That's why I like to put a 100K (or 1Meg) potentiometer in series with the 1K resistor so I can set the brightness low enough. An MSP430, or even an Arduino, should be able to set a digital potentiometer similarly to keep the brightness constant.
P.S.
This file was created in LTSpice, which I find simulates a Joule Thief fairly accurately and quickly. And I can play with circuits quickly, to learn a few things that way. But there are some differences, of course, between the simulations and reality, and I'm still finding those things out myself, not being a degreed EE.
Finally, if your supercap is below 0.5 V and the Joule Thief won't start up, you can put an AA, an AAA, or even a CR2032, in a diode-protected arrangement similar to what has been shown above (so that the CR2032 doesn't have all of its power go into the supercap), and you can put the AAA in parallel with the supercap with a momentary switch to start up the Joule Thief. Then, the above circuit should keep the Joule Thief going, even though the supercap is below 0.5 Volts.
In addition to bringing back the voltage from the LED's to achieve really low run voltages, I have also found that replacing the 1K resistor of the joule thief with a constant current source regulates output power fairly well.
And this also works with a high voltage JT (such as 6V into 5 white LED's), which are generally more efficient than the standard JT.
