life death finds a way
Look, there are a few things to consider here:
- Book vs Show canon, and differences therein
- Magic at the Wall
- The Night King's abilities
- Dragons and their connection with magic
- Trumpet time
Caveat: I will mix my answer with book and show canon, but will endeavour to stick to the confirmed show canon where possible
The Wall, more than just ice, ice, baby
The Wall was built approximately 8,000 years ago (if the legends are to be believed). It was built presumably to keep the Free Folk (known to Westerosi as 'Wildlings') out, but in reality was actually built to keep the Others (White Walkers) out. It was built by Brandon the Builder, with the help of the Children of the Forest, and seems to have magical properties.
In the show, I don't believe it is confirmed that the Wall has magical wards, but it definitely is in the books. In fact, in the scene where Sam helps Bran and co. to pass through the Wall, you can clearly see that there's no mention of the magical wards. Whereas, in the corresponding chapter from the books, Bran and his group actually need to pass through the 'Black Gate' below the Nightfort castle, and only a member of the Night's Watch can allow them to do so.
So this establishes that there is most likely some magical ward stopping the dead from passing through the Wall. It's important to mention that this same ward is believe to have been in place at the Cave of the Three-eyed Raven.
The Night King has the magic touch
Ok, so, now we move onto the abilities of the Night King, one of which seems to be that he can 'Mark' someone. This grants him the ability to know where the person marked is, and can get past the magical wards protecting them. As proven in the episode Hold the Door.
All the Night King would have to do is wait until he can 'mark' someone able to pass the Wall, and literally follow them in. Yes, in this particular case it does help to have undead giants able to smash through the physical barriers.
Dragons, White Walkers and Magic
Listen, all of this stuff is somehow connected, either by the fact that they are originating from the same source of magic, or that one way or another they can sense and affect each other.
It has been widely mentioned that the dragons have returned with magic, or is it the other way around? The Others seem to have returned with magic, or is that the other way around? It doesn't matter, which is my point: they are connected.
It wouldn't be too far-fetched to surmise that perhaps the Night King can feel the presence of dragons. All he'd have to do in that case is wait until one comes flying straight to him. Oh wait, he did do that!
The Night King is Horny
Lol, like forreal though. There exists (at least in legend) a horn believed to have the ability to bring down the Wall, the Horn of Joramun (aka the 'Horn of Winter'). Perhaps the Night King has, or can, or would find this horn.
The Night King is a resourceful dude, he has shown capability beyond imagination. He has waited for over 8,000 years. Something tells me that he's ready and there won't be anything to stop his progress.