Is it possible to get closer to stationarity by combining 2 of the same futures contract but for different expirations (calendar) to form a pairs trade?
Maybe, if you look hard enough. I can't think of an example that is guaranteed to take any of the [ts2,ts6] out of the equation. As traider said, it's akin to finding the holy grail.
Most futures calendar spreads behave similarly to the underlying (in terms of trending vs. mean reversion - the spread could move opposite the underlying however). Spread traders looking for mean-reverting trades generally construct butterfly or double-butterfly spreads. However, transaction costs are much higher and you can still get killed by a freak outlier. TL;DR. No free lunch, but look into butterfly futures spreads for your analysis.