Actually I've brought the matter up several times over the past few years. A series of home made units that started with powering my Purple Days in the garden. Since it's mains driven, the only other option is a long extension cord......
Here you can see the latest version in front of my 'Solo Road Case':
As you can see it has three 18650s in series and the right connector to mate up with the extension cable that fits the Solo. This, along with the charger, PA (my units can take advantage of this feature being older), stem, stash, Solo and so on go in the camera case with it (it goes in a side pocket). I suggest this scheme (three in series 'barefoot') to the much larger one you suggest (3 times bigger?). Extra capacity is limited of course by how many battery sets you're willing to buy, charge and carry.
You should check your math before making plans. Yes, Solo uses 2200 mAh batteries (actually properly called "cells", two or more of which make a battery.......) but it has
two in series. It needs something around 2200 mAh
at 12 Volts to recharge. You really only have a maximum of 5200 mAh, of which you can get to maybe 80 or 85%? Think about it, if you put those 6 cells, two at a time, into the Solo that would be 3 pairs, right? 3 'recharges', not "7.09". And, due to the fact that you can't use them at 7.2 Volts, but are forced to use them at 12 Volts through the charger (the extra 'battery power' going to heat during recharge) you really can only get two full recharges. Them's the rules.
Your phone will have about the same 'conversion efficiency', but the NiMH in the MFLB case will be somewhat worse. Useful, but not as much as you're thinking I think?
While I'm not sure what to think about the OP's charging issue (I'm not sure it's a real issue since the battery comes up fully charged?), you might want to rethink that charge procedure some. Many times I've suggested here considering disconnecting when the 7 light lights but before the flashing stops. Doing so will double or more your battery life. Consider the information around chart 4:
http://batteryuniversity.com/learn/article/how_to_prolong_lithium_based_batteries
I'm currently having some friends test my latest kluge to do this automatically. It seems to be working quite well, it looks like this:
The Solo charger plugs in upper left and the output (to the Solo) connects to the blue terminal block next to it. The 'start' button is lower right. When charging is cut off (at a level set by the screw above the switch) it beeps at you by the beeper under the washing cover to the left of the switch. Since it cuts off the much slower (and less efficient) part of the charge curve to the user it charges much faster (1/3 to 1/2 less time) and has very little if any useful 'sessions per charge'. And the battery lasts 2 or 3 times longer. You can get the same benefits manually but you have to watch it and guess a bit (since you really want to go 'a while' with the light still flashing for optimum results I think).
BTW when you get to the 'need the funny cable to mate up with the Solo' part you might want to contact Pipes, I think has some left over from his PA project. The connector, 1.7X4mm (AKA EIAJ-02) can also be bought from a few places (I order mine from Digikey) if you're game for making your own.
Fun idea, but not as popular as you might think?
Best wishes in your endeavors.
OF