Reach out to your undergraduate alumni clubs. Find out who runs the one in your city. If no one does, then start your own. The school will help.
Reach out to any existing or previous charities / groups / club you were a part of. Ask them if they have any issues or needs. Meet with them and ideally meet with a director of the club or charity. Offer to help, but ideally in a strategic area. That is, if you contact a puppy pound, ask them what they key challenges are, not how to volunteer on a saturday. Build rapport, get to know someone in the org, and integrate yourself in a role beyond that of "volunteer kennel cleaner".
Try coming out and outright asking for a significant role in a charity or - this works surprisingly well. Youd be shocked really. Seriously, try it. You wont believe the results.
Check out meetup.com - its INSANELY easy to join something local or start your own (running your own is not free, but in like two weeks I made a 30 person club - just pick something smart - ie not a jane austen club...)... I did this late, in the game and never even listed it on my apps cause I thought my more long standing stuff would seem more meaningful, but if you start early enough you can build something here.
If you were part of a frat, call the frat, talk to the president. Ask him how things are, bla bla, ask him if he needs help with anything. I know a guy who got some kind of "Director of Fund Raising" role by doing this. It's kind of bull, and that irritates me on a variety of levels, but he promises he'll actually do something at some point.