I'm surprised appropinquo (sometimes spelled adpropinquo) hasn't been mentioned yet. Caesar (Bellum Civile 3.9) also uses this one, and to my ears, it better fits what is being said in the show.
jamque hiems adpropinquabat
At that point winter was approaching...
I can envision Romans, worried about the coming winter, saying "Hiems appropinquat."
Visually, if you imagine winter to be a person, you could see that person approaching you, but they're not yet there. Meanwhile, subesse means more "at hand," where that person is standing near you but hasn't yet begun to speak; perhaps that person is just nearby.
Similarly, nox appropinquat would mean late afternoon, while nox subest would mean twilight: the sun has just set, but it's still somewhat light out.
This makes sense if we look at a few parallel passages. Livy (3.34) and Cicero both talk about the time relative to the day of the comitia.
A rumour was then spread that two tables were wanting; on the addition of which a body, as it were, of the whole Roman law might be completed. The expectation of this, as the day of election approached, created a desire to appoint decemvirs again.
volgatur deinde rumor duas deesse tabulas quibus adiectis absolvi posse velut corpus omnis Romani iuris. ea exspectatio, cum dies comitiorum adpropinquaret, desiderium decemviros iterum creandi fecit.
The election was fast approaching, but it had not yet arrived. Meanwhile, Cicero's parallel (Pro Milone 42) using subesse shows a more immediate closeness with less movement.
especially, O judges, when the day of contest for the greatest distinction of the state, and the day of the comitia, was at hand.
praesertim, iudices, cum honoris amplissimi contentio et dies comitiorum subesset.
The event being described as occurring when the comitia were being called together, not days before, as they were in the Livy passage. Though voting hadn't taken place it, the nearness of it was already established and not coming into nearness. It was at hand.
This makes sense even etymologically. The root implies non-movement: esse is simply a form of "to be." When Caesar (BG 1.25) expressed the relationship of a town to a mountain, he used subesse:
Tandem vulneribus defessi et pedem referre et, quod mons suberit circiter mille passuum spatio, eo se recipere coeperunt.
At length, worn out with wounds, they began to give way, and, as there was in the neighborhood a mountain about a mile off, to betake themselves thither.
The mountain isn't moving closer to them, but rather is described as close (only a mile off). If anyone were to be moving, it would be the soldiers, and the word to use here (of the two) would have been approprinquare.
We can again visualize the way the word works by comparing it to other esse words. Below I put TU et EGO together to show the relationship of the verbs:
1. TUEGO (adesse)
2. TU EGO (subesse)
4. TU EGO (abesse)
To put all this back into winter, bearing in mind that language can be imprecise, and bearing in mind that descriptions can be exaggerated (so I'm not saying necessarily that hiems subest is wrong!), but were I to hear hiems subest, I would imagine December already, on the eve of winter, whereas I would use hiems appropinquat to describe the action of winter approaching, though it still be fall. This could be late October when the chilly north wind blows or November when frost begins to appear. In either case, signs are present that winter is approaching, but not yet here.