I'd like to do that, but it's impossible with current Downloads API. I can't save files to any folder outside of your default save folder, and I can't open "Save As" dialog in your last saved folder because it might be outside of it as well.
And even IF we're talking only about the folders inside the default save path, which there's no way of knowing, it would only be possible with some really dirty hacks.

https://bugzilla.mozilla.org/show_bug.cgi?id=1342563
They're talking about possible "useLastDir" option, that's exactly what you want. I'll add this feature the moment Mozilla implement it.
Thanks for the feedback.