I don't think this rise is "the rise". We'll probably still have many months of little price movement, as we've seen the last couple months, a kind of staircase shape till the thing goes parabolic.
The last halving was priced in by the time it happened. The slow steady rise we have seen is probably the halving being slowly priced in long before it happens. I think you're right about expecting lots of little price movements pushing the price steadily upwards until the halving. There will be little ups and downs, but overall a steady price increase.