I would imagine there is enough variety in different program's practices that any small advantage in this regard is difficult to predict. It's hard to say for each program when "early" and "late" is. Some start earlier than others, while other programs end late into January. I also think these programs understand that the schedule of 4th years is widely variable, such that you may be on vacation in January, or November, thus your available interview dates may be limited.
I know this is the time of year when this board lights up with anxiety over interviews, and certainly some of it is founded. I don't think the timing of your interview dates should should be planned to give a subtle message to the program about your interest level, nor do I think programs interpret it as such. You will no doubt hear endless stories of people going on that last interview in mid-January, not really caring anymore, but eventually falling in love with the program and matching.
Final advice: control what you can, but schedule the interviews for convenience and economy, not to send a message about your dedication or interest. Try to decrease the number of things you stress over. It really will be a benign process when it's all over.