For men who are recognizing that social isolation is a long term risk in their lives and want to do something about it, I heartily recommend looking into these organizations. They're a mixed bag, like anything, but a bunch of guys who can hold anything together for decades should make anyone's list of options to investigate.
I recently became an Odd Fellow and I still feel the same with respect to whether a member is gay or not. The big difference with the Odd Fellows is that both men and women can be (and are) members. The Masonic orders are only for men.
In any case, I wouldn't write them off completely due to your concern.
Being recognized as a good guy by a bunch of other guys is no small feat. Someone who is trustworthy, kind, fair minded, honest, with only moderate conceits, good natured, charitable, and magnanimous is rare to begin with. I think if someone has these qualities, there is very, very little that will stand in their way.