@dlowan,
I've only gotten it when flying east, say California to Italy. I do remember that the last time I flew home, I slept for something like thirty hours off and on. (That must have been a weekend.)
Years ago I read a book about jet lag and remember the advice to avoid coffee for 24 hrs before the flight, on the flight/flights, and then when landing (I presume they meant in the daytime), drinking several cups of strong coffee. The book made a point that it wasn't caffeine that mattered, but something else in coffee.
My first trip was totally screwed up (35 or so hours awake) and no coffee trick mattered; second trip, I think we arrived late in the daytime and thus sleeping (I sleep little on planes, alas) made perfect sense and I had little jet lag. Third trip, arrived in the daytime, wandered around like a zombie but did have several coffees after landing, slept well that night. So, I don't know if that coffee thing was silly or not.
I suppose there is advice online that is more up to date.
I never tried melatonin, though I bought some - something made me chicken to try it.