Besides if you're trying to restore humidity lost by heating your room then pure energy wise there's not that much difference in evaporating the required amount of water using ambient heat or evaporating the required amount of water using a different energy source. The only thing you'll achieve by using ambient heat is that your room will cool by the amount of heat required to evaporate the water (a good thing in the summer, somewhat pointless if your aim was to make a room both warm and comfortably humid).
Fundamental thermodynamics will probably mean the 'wait and see' approach is more efficient in the end, but it makes no promises on efficacy. The difference in efficiency probably lies in the fact that you can heat a room with an efficiency >100% if done right, if you haven't managed that yet it probably doesn't really matter which way you humidify the room.