Currently deers in HnH are not very good, as we have seasons, they need seasonality.
First of all during the late fall, female deers (hinds) produce less some specific hormone, which results in drop of testosterone levels in male deers(stags), and they shed their antlers!
Also, only stags have antlers, currently all deers in a group have antlers. Which is just wrong!
Deers actually form harems during mating season(rut), where there is 1 stag deer and then from few to dozen hinds.
Sometimes there may be more than 1 stag in a harem, if food is plentiful.
Thus I propose the following changes:
Group behaviour
During winter, spring and summer, deers often move in single-sex groups, so there should be groups with only stag (with antlers) and only groups with hinds (without antlers).
During fall, deers should form harems, with only one stag in them and then multiple hinds. During fall there should also be weak beta males that don't have harems in small groups/alone.
Antlers
During winter, elks shouldn't have any antlers at all.
In the spring, they should gradually start to have antlers more often,.
During the summer, and fall they should always have antlers.
You could perhaps sometimes find shed antlers from the forest during late fall/early winter.
Battles
During the rut (which is in fall) stags should sometimes fight between each other, often only to show off, but sometimes to death.
I think that these changes would be relatively easy to do, and really add more immersion and meaningfulness for the seasons.
Other animals could also get some changes depending on the season, but that topic is to be discussed in another thread.