Stiny hay might be getting too much water or shade.  It isn't generally stinky.
It should be just wet enough to feel like a very slightly damp sponge when you handle it.  
If there's mold in it, it's generally too dry.  If there's runoff, too wet.
I might move my compost bin if it's in the shade.
Compost generally doesn't smell bad, if it's mixed up properly - if it's ALL hay it might not break down well.  If it's not getting turned every couple months, same.
I mix my hay with - well - however much corn cob bedding comes out of the stall with the manure and urine.
I don't put any additives on it, lime or DE would tend to make it inhospitable to the creatures it needs.  But I do turn it about once a month.
If I can, I try to run over whatever I put in the pile, with the mower or tractor mower, so it's in smaller pieces.