What's wild is that with a dragon the Night King could go zipping off anywhere in Westeros and easily start a second army from scratch. Which means he probably won't do that.
excellent point -- provided that the dragon didn't incinerate the people it killed which is what happened to everyone else the dragons burned.
While he could start a separate army of wights, it wouldn't result in a growing army unless he stuck around for an extended period. Wherever the night king is, it's like a snowball rolling down a hill gaining size based upon how many people he rolls over. Once he leaves, the 'raise dead' ability would likely leave that army too. Granted, the remaining wights could certainly wreck havoc on wherever they are.
It'd probably be too risky for him to fly directly into kings landing alone, but that would be a fertile city to start an army. Probably his best move would be to fly into a corner of King's Landing with a few white walker generals and start from there. It's fun to theorize if you were him.
Normally I get a bit irked by stupid character behavior, but with the night king I'm not too bothered b/c he represents death - a slow and unstoppable lineal force. Death doesn't strategize, it just keeps on rolling.