when you get settled into your job, you can find yourself with a lot of free time. it's important to have hobbies here, otherwise you turn into one of the expats who smoke and drink all the time. people i know study korean, work on a masters degree, play music, read, write, knit, draw, paint, etc.
i was on the hunt for Kyeong Saeng Won orphanage to volunteer there b/c it's in jamsil where i live, but i havent found it yet. in my search, i found this website that helps expats find places to volunteer. Thier main focus is on English lessons for children from low-income or disadvantaged families in Seoul, but they also do things like serve food to the homeless and help out at orphanages.
it's untrue that youre not allowed to volunteer under an e-2 visa, btw. so long as you arent getting paid, you can volunteer where ever you want. i should mention i first heard about HOPE via my coworker xohoua at cdi pyeongchon. http://www.alwayshope.or.kr/