The skin on your face is a lot more sensitive on the skin on your hands, so it’s more likely to react. My girlfriend had a reaction a lot like yours on her cheeks when she tried a new lotion. She’d rubbed it all over her face, but it was still just the cheeks that broke out.
You shouldn’t be using any lotions on treated skin as mentioned in this post previously.
Out of tea tree oil, witch hazel and aloe vera (I hope you’re using the pure clear stuff, not lotions with it), I would probably be most hesitant about tea tree oil as it’s probably most likely to cause an allergic reaction (it’s pretty potent). I would use as little as possible next time and only have a 15-min treatment or so, and see how that goes.
Although there is a small population that is sensitive to tea tree oil, most reactions are from people using too much. A little goes a long way, but people want to pour out the whole bottle thinking that they will get better faster results. All they will get for that waste of product is dry flakey, red irritated skin.
Another factor with tea tree oils is the purity and the grade of oil. Some lower grade oils are more harsh, and therefore more prone to irritation. Pharmaceutical Grade Tea Tree Oil is the lowest grade one should use for post electrolysis treatment.