Here I have two Li-Ion cells ready to solder together into a parallel pack. The technique that I found to work best was to insert a coil of small electronic solder between the tab and the bottom of the next battery, making sure that the solder never overlapped, so that their was always only one thickness of solder thick at any particular place. Then, I'd hold down the end of the tab with a screwdriver (to make sure it didn't bounce back up while the solder was still hot) and heat the tab with heavy downward pressure from a heavy duity (stained glass) soldering iron for a few seconds. I'd press down until the solder visiblely melted (usually accompanied by a puff of resin smoke) and the tab settled down. Then I'd remove the iron while keeping pressure with the screwdriver.
Important safety note: At the factory, tabs are welded to li-ion cells very quickly. Li-Ion cells will catch fire explosively if they are overheated. It is very important that you do not leave a soldering iron on the cells long enough to raise their internal temperatures to dangerous levels. Heat the tab, melt the solder, and then remove the iron.
You will note that I used one set of tabs to attach the two cells, and the other set of tabs was positioned 90 degrees off axis, to be used to connect to other packs of cells. Note that I'd use tape to hold the cells closely together while soldering them, but would remove the tape later to make them fit into the battery holder.
Here is my first pack, soldered to the charging electronics. Note that I have temporarily covered the exposed positive ends of these cells with clear plastic tape. This tape insulation is very important! It keeps this pack of cells from accidentally shorting through the pack of cells that I will add next. The process involves laying the cells on top of each other while soldering, and it is very dangerous to short out li-ion cells. (All of the charging electronics and fuses in the battery are designed just to keep that from happening.) Note that other things on your workbench can short out li-ion cells. For example, a coil of solder conducts electricity very well, at least until it melts. So do screwdrivers and pliers, and they are less likely to melt. Li-Ion cells are usually stored and shipped 40% full to prolong their life, and the ones I had certainly did spark a few times. A small spark here or there won't hurt anything, but if you short a cell for longer than a moment, it will begin to overheat and you could get into a runaway situation where it eventually catches fire explosively. Don't do that!
Once I've safely insulated the ends of these cells, I add the 2nd pack and solder them together.
After the tabs are soldered together, you can unfold the 2nd pack and lay it down flat.
When soldering existing metal tabs from the charging electronics to the cells, I found it best to coat the tabs with solder first, and then attach them to the cells. Below you can see a pre-existing tab that I pre-soldered before attempting to attach it to the back of the cells.
Note that the first and second pack have a good space between them (by design) in the battery holder, which leaves plenty of room for the solder tabs to be folded between the two packs of cells. However, the second and third packs must be mounted much closer together. Because I did not have a custom manufactured set of solder tabs, I found it easier to just butt the batteries up against each other and join the solder tabs above the cells.
Now, I have all three packs connected in serial, and to the charging electronics. All that remains is to find some very sticky double-sided tape to affix everything to the interior of the battery case. I suggest you use carpet-tape (designed for tacking down carpet), which appears to be about as strong as the original tape I removed from the case when disassembling it.
Once the tape is in place, all that remains is to snap the two halves of the battery case together, being careful to get all of the tabs in the right slots. Everything snaps together with a satisfying "clicking" noise, and the battery is as good as new. If you do not remove the manufacturers sticker (as I did) and are careful to use a plastic tool and not leave scrape marks when prying the battery apart (as, once again, I did) the battery will look practically like new.
But how does our existing charging charging circuitry deal with the brand new li-ion cells?