Gr33kfan,
My daughter was a social chair for her sorority last year, so I'll give you her input (she's currently enjoying being a newly graduated lazy bum and doesn't want to type anything again ever at the moment.
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This is just her personal preference, but she said she didn't want phone calls or emails- she wanted texts. If she knew that for some reason, her sorority didn't want to do something with a particular fraternity (popularity of the fraternity, by the way, wasn't usually the reason for this), or couldn't do something because of timing issues, it was easier for her to deal with that via text. Phone calls where she had to say no she found awkward and texts prevented that awkwardness. For that same reason, she wasn't hyped on getting together for coffee, etc although she did it from time to time with some social chairs (usually after she had already said yes to an event and they needed to plan).
She said persistence is good, but there is a fine line between persistance and desperate - sort of the same line that would exist with dating. She said she personally wouldn't be super comfortable with someone showing up at the door and in her case it wouldn't have been effective since she didn't live in the house most of the time she was social chair.
She did agree with a lot of other suggestions that have been made so far and added a few.
1) Yes, go and serenade at pre-chapter dinners. While you're there, the social chair should introduce himself, leave his number, and suggest that you'd love to do something with them.
2) DON"T leave it up to the sororities to plan something. Call them with a plan already made or offer to help plan something together. Do something in conjunction with another fraternity.
If you have a mixer or unofficial - and they are hosting at one of their places - offer to stay and help clean up. She was so impressed with a new fraternity on their campus that came back the next morning to help pick up her apartment after a mixer. Yes, she put them on the schedule again - quickly.
3) Show up to philanthropies, make sure you are there with big numbers and get involved in them. If you participate in something like, for instance, a Mr. Greek contest - go all out and REALLY work at it. Impress!
4) Ask them over for dinner.
5) Have an BIG event at the start of the year - dance, BBQ, whatever and invite EVERYONE(okay, not literally everyone) - campus wide. It's a good rush event for you, it helps sororities to get to know you in a comfortable atomsphere, and you get to know girls that might be rushing before they get into a sorority and get "locked in" to certain fraternities their sorority might hang out with. If those new girls like you - they will start pushing their own social chair to do stuff with you.
One of the fraternities on her campus also has helped freshmen move in every year. Yeah, they focused mostly on helping the girls. Because of this, she liked them from the start and remembered them when she went to plan functions. That same group planned a campus-wide flag football tournament the year before that a bunch of sororities participated in.
6) Yes, invite groups/organizations over that aren't sororities. A fraternity at her first campus invited the cheerleaders over for dinner (before she had joined a sorority). She went back to hang out several times after that and as she met sorority girls throughout the year, she had good things to say about that group. Also, two of the girls on her team were already in sororities and THEY had good things to say about that group to their sisters.
7) Bring over cookies, food, send cards etc. One of the fairly new fraternities on her campus sent candy and a card to every sorority on Valentine's Day.
8) The minute I read to her the part about a lot of the guys in your fraternity having girlfriends, she said, "Yeah, and that's part of the problem, probably."
Her experience has been that if a lot of the guys in a fraternity are in relationship they either aren't too interested in socials/mixer or they don't show up for them or if they do show up they stand by the wall. Because of this, she suggests focusing on your freshman/sophomore guys who are less likely to be in relationships and make sure everyone who comes is motivated to show up and socialize (and you're right - this doesn't mean hooking up. It just means be friendly and have fun).
Hope that helps.
ETA: Oh my goodness, this ended up being way longer than I intended - apologies.