In a volatile market with uncertainty it's more likely that we see a volatility skew and not so much a smile. Therefore it must hold that, in chaotic markets, out-of-the-money calls and in-the-money puts are in greater demand than in-the-money calls and out-of-the-money puts.(am I correct?)
But how come OTM calls and ITM puts are more in demand in volatile markets?
Eg. the day after Brexit there is a skew for GBPUSD options
Actually, you are going to find that the vol premium on the wings is lower in volatile markets, purely due to the "cabinet effect" when vol is very low.
I think, however, that you are mixing up smile (aka kurtosis premium) with skew (risk direction premium). When a specific side of the vol surface moves that usually means people are trying to protect against a specific direction move, either due to perception or due to market position
