This applies primarily to a solid/liquid system. It is straightforward to get a well equilibrated crystalline solid: you take the crystal structure and keep it at the desired temperature for a bit.
To turn half of that system into a liquid, however, takes much more effort. Melting is an activated process, so there can be a substantial hysteresis to the melting, i.e. you need to heat to a much higher temperature than the melting point to get to the liquid state. In addition, you need to eliminate all “memory” of the ordered crystalline state. That is most easily achieved by picking an even higher temperature and keep the system at that temperature for long enough before quenching it down to the melting point again. Because of the hysteresis and also crystallization being an activated process, you could even go below the melting point, which would give you a supercooled liquid (it even works for macroscopic systems. You ever seen the experiment where you instantly freeze sparkling water by taking a bottle out of the freezer and opening it?). But you really want to equilibrate it at the estimated melting point temperature for the coexistence simulation.
Using NPT should not be necessary and is counterproductive. NVT should do and make things easier. You don’t want the system to be at the density of the higher temperature, but the density of the melting point. Letting the system expand just makes it harder to compress it later. It is much easier to expand a liquid system than to contract it.