The criteria for this is actually pretty straight-forward:
1. Anything in a module will not carry thru to the individual emails. As far as I know there's a kind of separation between the bay of modules on the right side of the editor (which will get updated) and the actual modules that are drag-n-dropped into the canvas to create an email layout (which will not get updated).
2. Anything that's not in a module and instead hard-coded into the template will change once the template and email asset are reapproved.
In your case, if you changed the social icon at the template level and that wasn't inherited thru to the emails once you re-approved them, it'd suggest to me that you're editing the social image within the context of a module.
CHEAP TRICKS:
Let's say you've got a case where you want to replace the old bird logo for Twitter with the new X icon, since this one has been coming up more recently. In that case, you might have an image uploaded to your design studio called something like:
icon-social-twitter.png
One way to think about this change would be to swap out the image above w/ something like "icon-social-x.png" but that would mean that you'd have to go into each email asset and remove the old module in the canvas (which doesn't update) and add in the new module from the module bay (which does update).
Instead, you could use the "Replace File/Image" feature and load the new "X" icon in place of the old bird icon but keep the file name the same. In this case, you'd name your new image "icon-social-twitter.png" (or whatever the existing filename for the image is) and make sure it was the exact size (height/width) as the existing image. Then you'd go into Images/Files folder and find the existing file called "icon-social-twitter.png" and use the "Replace Image/File" feature to load the new file in place of the old file. In this way, you only update the file and nothing on the template needs to change so it'll be a seamless transition without any of the overhead. This works b/c the emails and templates are still calling to the same file using the same name, but now there's a different image in that place so nothing else needs to be updated.