So I've played around with this a bit...and I'm not sure if the alternative (i.e. keeping track of the current music time and returning to it) is actually better, since when you die the music changes to the first course character so you don't get continuity either way and it feels disjointed (especially if you die quickly).
Plus, one of the draws of the current system is that you can tell how fast you're going through the level via audio cue (i.e. which part of the song you're up to tells you what pace you're on). Maintaining that requires that the music time be set to 0 when the game time is, so doing anything to maintain the position in the song would remove it.
(Note that in practice mode, where the timer doesn't reset on death and the audio cue thing is irrelevant, the music actually doesn't reset when you die. Additionally I'm planning to add at least one other mode where the timer keeps going on death and this will use the full soundtrack in the same way as well)
I did try several other options, but I didn't think any of them were an improvement over the status quo. If there was another solution you were thinking of, I can potentially give that a go instead. (For after the jam is finished, of course :v)