I've found that the best way to be more empathetic is to become a calmer more patient person. Putting myself in someones frame of mind is like taking a mental deep breath after every time they say or do something. I have to let the loop of consciousness run a few more times before allowing my thoughts to become opinions and express them in a response.
I recommend meditation. Empathy is consciousness of self and of others, consciousness is pre-frontal cortex executive function. Meditation is the expansion of self-awareness. When I was younger, sometimes after getting angry, like reeaally angry, seconds after calming down I would think "Shit, there was no reason to get that mad". It's a complete loss of self-control, executive function. The same thing happens after saying something stupid(hurtful) and then realizing it was hurtful(not empathy), if I have the ability to realize I shouldn't have said it, that means if I had just mentally waited a second or two and reassessed the thought I could have prevented myself from saying it. Meditation helps give me this time, stay calm, and assess situations(not only social) before acting. I think if scientists(if they haven't already) looked at the brains of empathetic people and stay calm in chaotic situations they would find the front of their brain buzzing with activity.