I haven't been to The Red, but I was bummed you didn't get any responses, so I thought I'd tell you what I'd do in your situation.
Usually when I'm climbing somewhere I haven't been, I look up local climbing and see if they have a Facebook group or a forum, and then look for rides there. Sometimes that stuff isn't immediately obvious, so I'll contact a local gym to ask for information. In this case, LEF Climbing in Lexington may be your best bet. I'd call them and see if there are any local outdoor groups you could get in contact with.
Just be sure to support any local businesses you get help from! For example, when I stayed in Chatt, the folks at High Point were really helpful, so I got a month membership and took a class with Arno Ilgner while I was there. I probably could have found a cheaper way to get showers while living out of a car, but I don't mind supporting the local climbing communities and I got a lot more out of the rock gym than I would have gotten out of a Planet Fitness or whatever.