There's currently no way of showing the popup again. I think it could be tricky to find a good way to add a button that is at the same time not in the way of other GUI components and makes it clear what it does. An alternative could be to not have the popup disappear automatically if the user interacts with it (click, mouse over etc.)

Thanks or maybe if possible access it from the version list....

My slow brain reacts when the popup has disappeared and would like to look at it again ;-)
Lesson learned is if you see be prepared to read it ;-)

@Sebastian_Berlin-WMSE It's possible to make the script add a portlet link to the toolbox, and have that link .show() the dialog again. The portlet link is added when the dialog closes (both when the user closes it as well as from the setTimeout). This guarantees that there will always be a dialog to show when clicking this link. It also guarantees that the portlet link is only shown after the user has made an edit.
Proposed diff: https://sv.wikipedia.org/w/index.php?oldid=46879235&diff=46881200
If you think that this is a good idea, I can implement this.

Looks mostly good. A few things:

1. Adding and removing the portlet link seems like odd behaviour to me. If nothing else it shifts the toolbar around. I'd have the link added at load and just not do anything if the popup is visible.
2. If you show the popup after it's been hidden it won't go away by itself. This is reasonable, but it means that the user needs to click the "More" button in order to get the "Close" button. I'd add the "Close" button in the non-expanded view.
3. You don't need to use find() in close(). $( '#gadget-articleQuality a' ) gets you the link.

In T240228#5911117, @Sebastian_Berlin-WMSE wrote:

Looks mostly good. A few things:

1. Adding and removing the portlet link seems like odd behaviour to me. If nothing else it shifts the toolbar around. I'd have the link added at load and just not do anything if the popup is visible.
2. If you show the popup after it's been hidden it won't go away by itself. This is reasonable, but it means that the user needs to click the "More" button in order to get the "Close" button. I'd add the "Close" button in the non-expanded view.
3. You don't need to use find() in close(). $( '#gadget-articleQuality a' ) gets you the link.

@Sebastian_Berlin-WMSE @Nirmos is this an implementation that still looks reasonable or has the need changed since this was first discussed?

I think it is still reasonable and an improvement. I haven't seen any followup on my review.