Something I noticed in the JOLTS data was that if you subtracted out a "dynamic equilibrium" (log-linear path) , the residual data was almost sinusoidal (click to enlarge):
I wouldn't expect this sinusoidal fluctuation with a constant frequency to continue simply because there aren't a lot of true waves in economics — the sine wave is more of a curiosity [ETA: the period is about 3.5 years].
But then I noticed that this pattern matched the same residual S&P 500 fairly well, but with a lag of about a year (meaning the S&P predicts the fluctuation):
In fact, all of the other JOLTS data series appear to show this correlation as well (separations = TSR, hires = HIR, quits = QUR):
Even the unemployment rate shows the same fluctuation — but with a lead of about 16 months:
That is to say this would indicate the fluctuations in the unemployment rate predict the S&P 500. As the unemployment data goes back farther, I can look at whether the correlation holds up over time. Eyeballing the data, it actually looks more correlated at zero lag/lead. The run-up in the market before the 1987 crash (might be interesting viewed in the context of the post-80s recession step response), the dip in the mid-90s, and the dip in the 2002-2003 time period line up with fluctuations of the unemployment rate data away from a local log-linear fit. It's also just much noisier in general.
In any case, I will look more closely at this as well as "help wanted" index data from Barnichon (2010) (see e.g. here).
Update 14 September 2019
One thing I would like to point out is that this fluctuation is on the order of a few percent on top of the dynamic equilibrium path — a second order effect.
 Take the data series JTSJOR (job openings on FRED) take the log and fit a line y = a t + b to the log data and subtract log JTSJOR − (a t + b) ≡ Δlog JTSJOR.