Why does exhaust smells linger some days? Some days I can smell bad smelling exhaust 50 meters or more away from the bus stop. I think the air is usually moist those days. 

Does exhaust gases linger longer if the air is moist near the ground? Is it smog I am smelling those days?

On morning 12 march it was about


Temperature: 4°C
Humidity:93%
Barometer:978 mbar
Wind:10 m/s


Looking at the data from earth.nullschool.net at a few hours before that morning I saw surface temperature of 5.1°C and 5.8°C at 1000hPa.
You can see this post for a more detailed discussion, but the typical answer to your question is air stagnation.  When the atmosphere is stable, there is little mixing, and the air can stagnate.  Typically this coincides with low wind speeds and a low boundary layer.  The boundary layer holds air near the surface like a blanket. The boundary layer collapses overnight, becoming shallow, and grows when the sun heats the surface of the Earth.  So, in the morning, the shallow boundary layer allows less vertical mixing to occur.

You might be noticing that on cloudy days, when clouds are low, there is little wind and no solar heating.  Air pollution has no place to go, so it sticks around.  

Another aspect is that vehicles emit more emissions under cold-start conditions.  This is because vehicles get cold overnight and take a while to get up to normal temperature when they startup in the morning.  The cold vehicle will have a low combustion efficiency and catalytic converters won't work as well.