As far as I know, there is no way to do it officially through the settings (checked in VLC 2.2.3).
There are a few alternative solutions, downside is none of them are perfect:
1 – Switch the "Continue playback" option to "Ask"
Problem with this is if you miss the notification to 'Continue' where you left off, then you'll have to manually search for where you were.
Here's how to change the setting:
Tools > Preferences > Click 'All'
Interface > Interface Options > Qt > Continue Playback Dropdown Options
Here's how the notification looks like:
It shows for about 5 seconds. While it is showing it would have already started the file from the beginning (so it won't 'wait' until either it gets a click or until five seconds have passed).
2 – Use another player for Music
Not a comfortable solution if you're so used to using VLC. The bright side is there are lots of good music players out there, so you might find one you like better than VLC.
According to a Lifehacker article, MusicBee is pretty good. I know Winamp was pretty good, but they shutdown a while back.
Side note: There could be other ways to do it unofficially. Perhaps modifying the source code.