it sounds like to me that 早めに＝ちょっと(いつもより)早く and 早く＝(いつもよりまた"早めに"より)早く.

「め」in 早め（はやめ）、大きめ（おおきめ）、多め（おおめ）,　少なめ（すくなめ）connotes a certain characteristic or tendency. As such, it has an effect of blurring the explicit meaning of the related words "early, large, many, few", etc.　 Depending on the context,「早め」could mean many different things, for example: "if anything, earlier," "on the early side," "somewhat early," "earlier than the usual or predetermined time," "as early/soon as possible," "early, allowing enough time (for something else)" etc. By the way, in the Kansai region, many people pronounce the words listed above as: 早いめ（はやいめ）、大きいめ（おおきいめ）、多いめ（おおいめ）,　少ないめ（すくないめ）.