I like the idea of a sort of "call and response" thing from different classes of instrument to mix it up and add light and shade. Banjo plays a phrase, squeezebox answers, fiddles have their say, whistles chip in an answer. I'm not saying that would be easy to call in a busy session but it might keep players on their toes - and keep the interest up for musicians as you're always in and out but you're not always starting at the same place in a verse.

An alternative might be calling random groups for each phrase...dog owners, back row, lager drinkers, the english, the irish, teachers, sexual deviants...etc...so long as SOMEONE keeps playing.

Or just pointing at rough segments/quarters of the room in order and see what sounds come out. That way people can anticipate their turn coming which might be less chaotic.

Perhaps best tried with better-known tunes that have become almost over-familiar.

I should say, I'm not really a session player so I don't know what session players would think of this, but as a listener to sessions I sometimes wish there was more variation and surprises.