The structure I use on longer trips: fix the ends, loosen the middle.
First two nights booked, last two nights booked, and the flights that bracket the whole thing. Everything between is booked a few days ahead as I go.
Why the ends specifically: arrival is when you are most tired and least capable, and departure is when a mistake is most expensive. The middle is when you are acclimatised and can handle a change of plan.
Booking a few days ahead rather than on the day is the practical middle. You get the choice and the prices without committing three weeks out, and you avoid the genuinely stressful version where you are looking for a room at eight in the evening.
Two things that make this work in practice:
Have a rough shape, not a schedule. These are the four places, roughly this order, roughly this long. That is enough to keep you moving and loose enough to change.
Book refundable where the price difference is small. Often it is a few percent, and it converts a fixed plan into an option. Where it is a large difference, that is usually a signal the place is in demand and should be booked.
On your current trip: check the cancellation terms on the middle nights. You may be more flexible than you think.