The way the binary .GIF format stores the image data
is somewhat involved, as with almost any image format and certainly any image format that can store multiple layers/frames/overlays, and I won't send you off to learn the fine details of extension blocks and the compression method. It won't helo you ag all
But GIMP and (probably, not that I have used any recent version of it) Photoshop lets you edit such layers, with their transparency and such, so you are left with little more than being arty with the on-screen artwork to get to grips with.
The most awkward thing is that a GIF that is optimised for transitiions may only have the very first image in the sequence as recognisably an image, with subsequent frames(/layers) being just the differences in the pixels fhat change, transparent in all those that don't. To back-form to the 'unoptimised' version, just hide all but the first two frames (first frame the full picture, second frame the overlay that changes things), copy all that is 'visible', paste that over frame 2 to make it explicitly everything, unhide frame 3 (overlays changes to the frame 2 situation), copy all visible, past onto 3. Or there's an actual "unoptimise" filter in GIMP that does this for yiu, but I don't think I've ever used it myself...
Then 'simply' edit each layer for new details (however you want it... I find that apt use of one or more separate image 'projects' can help organise anything you want pasting onto the working-project's frames (familiarity and practice and a bit of patience help, of which I have usually have two, but it's not actual magic).
If it's an animated GIF that you are editing, you probably don't need to do anything else when exporting to the new image (each time you want to see what your latest edit just did, perhaps) but you
can probably tweak the timings if you want, and it should be obvious how.
Once you're reasonably happy, save as-is (for when you want to come back and edit it a bit more) then re-optimise if you care about the file-size (or even if you don't, it's still good practice to) by the inbuilt capability of the editor (as with unoptimising, it's under Filter/Animation on the version of GIMP that I'm most familiar with).
I encourage everyone to
try doing something like this, in an idle moment. It can be fun if it's not work, and if you find out that you aren't too discombobulated by the process then maybe (one day, if not immediately) if it
does becomd part of your work then it is still fun rather than a horribly forced learning process to a tight deadline,.
. If you don't immediately understand it, then having no work pressure or immediate need to press on with it might help you ease into it.
(Also, this is one of few forums where I haven't gotten around to making myself an avatar, often an aniGif where that's allowed. More for want of any great inspiration or decisiveness, than anything else. Also I'd need to choose a decent third-parry host to upload it to and then point at.)
If I get time at the weekend, back home (or wherever I find myself with enough time and a handy machind to work on), I
will have a look at what you want. (And then I'd need to look for that hosting site, as well, so maybe I'll even satisfy my own lack, if I have extra spare moments.)