Okay. I think I see what's happened. Difficult to help remotely - but your explanations and pictures are helping a lot!
The spring obviously has two ends. One (still attached) goes to the Inner-Spindle of the Spool. The other end is the problem? It has to be secured. Otherwise the spring would just turn and turn.
These often slide over a POST or fit into a SLOT to secure them. Then the spring has something to pull against as it coils-up and tension builds.
In your first posting, Picture-2 shows FOUR posts. Assume we number them left-to-right 1,2,3,4. I'd say Post-1 or Post-4 look like the ones that stray end fits over/into. My money's on Post-4. It has the room around it for the thick metal.
I'd try prying up the end highlighted by the arrow and hooking it over Post-4.
Bent needle-nose pliers would grab it and allow you to manoeuvre it. All while keeping the remainder of the spring trapped by the spool. You can usually pull these around a fair bit (within their elastic-limit) and they retain their original shape.
When I tried describing discs, I meant something like a circle of hard-flat material that would slide between Spool and Spring-coil to prevent the coil from suddenly jumping apart. In your latest picture, it looks as though the spring is wanting to escape already, and you've only lifted the spool up a little bit. But if you don't have any tension on the coil yet, don't worry too much about restraining it any more than you are.
(BTW: You're not wearing your gloves in that picture)
