How can we better support others facing self-harm in their daily lives?

You said it right now, just be there to support them through it. Listen when you can, and do not judge. If you think they are in danger, reach out for help. They might be mad initially, but they’ll thank you down the road.

As someone who has self harmed myself, what I've found most helpful has been when people create an atmosphere of non-judgement. I often feel like a burden or disappointment after self harming and just having someone around who doesn't make me feel like I'm wasting their time is amazing. It's important to remember that everyone is different and has different things that help them - don't be afraid to ask the individual what they'd like you to do to best support them. Do some research into why people self harm and try to understand that it goes beyond just "attention-seeking". You don't need to be able to relate to the behavior in order to empathize with their emotions and so try to help support them with their underlying feelings. Generally, self harmer's don't like it when people "freak out" about self harm so just try to respond calmly and remember they're still the same person they were before you knew they self-harmed.

I think being understanding is the most important thing you can do for someone who self harms. It is important to focus on the emotions and feelings that are causing the self harm, rather than the self harm itself. Another important thing to remember is to not interrogate them about their self harm or make them show you any scars/injuries they may have as this can be quite a sensitive thing for someone. Also make sure not to make them feel bad for self harming, remember they will probably already be feeling really bad about it themselves and don't need extra pressure.