I have been pushing it into a pile since October. The sheep have a hay bale in a rack and eat off that, but when they are done with a bale, I push the remainder up into a pile and grab a new bale. There is a lot there because sheep waste a lot of hay, but when they get bored they will walk over and nibble on the better parts of the hay. By November it starts steaming, enough so that any snow on top of it starts melting, so it gets pretty warm fairly quick.
You are right though, I did add more hay to it yesterday. I shoved all the snow off the HUA, being careful to shove the snow around the hay rack first, then I shoved the hay into the pile. I normally would have waited a day or so for the sheep to chew on the bale more, but with so much snow on top of it, and the sheep about ready to lamb, I did not them pawing through snow to get something to eat. So I shoved a pretty big charge of it into the pile and just gave them a fresh bale.
No one will call the fire department though, everyone around me is family, and anyone driving by just wouldn't care. I clear a lot of land and burn a lot of brush piles every year and the fire department has never showed up yet, and I have had some brush piles burning that you could see from space!