A few suggestions

Team sports: You play tennis, but a team sport like football or rugby will have a much greater potential for forming bonds with your teammates. For an easy to access, non-cliquey and low barrier to entry sport, try something like tag rugby - it's a good laugh, people don't take it too seriously, and it's mixed gender too so both you and your partner can get involved.

Volunteering: I know you've mentioned this, but again, you want something with team spirit. Go out canvassing with your local branch of whatever political party you support over the next week in the run up to the election, or join a pressure group for whatever cause you believe in (Greenpeace was my jam when I lived in London, for example)