How do you make connections to faculty outside your department?
I organized a beer night for tenure-track profs in our department (and this was later expanded to related departments). the basic rules were: it's once a month, anyone is welcome, and it happens even if I'm the only one at the bar :). 

Do you have one person from a different department you can start this up with ? set up a mailing list and ask around to get maybe another 1-2 people involved. Start doing this regularly, and never lose a chance to mention this when talking to professors. 

Word will spread. Tenure-track profs need support groups. Our beer night is very popular :).