I just put the mouthpiece in my mouth every few minutes for whatever horn I'm not playing. Not sure how manageable cling wrap would be since outdoor gigs can also be windy.Yes, that's what I think as well.
On many mouthpiece caps there are a big hole in the top of the cap. Why? And they are also oversized. Probably so they are not damage the tip of the reed.
The best way to avoid reeds from drying out on stage between songs is to use cling film/plastic wrap. That would probaly help from drying out a reed.
The player can also drink water between the songs. That helps up.
I know a player that use 3-4 reeds during a gig. Some reeds that are soaking water.
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As for your 3-4 reeds/gig guy, I can imagine our conversation, "My reeds usually last about a month, how about yours?" "Oh, about 15 minutes. And I like 'em nice and water logged too." Then during the show, at the start of every tune, "Just give me a minute, I have to change reeds", as the band/audience leaves.
After reading this thread, I've decided to cancel my son's car insurance. I just have to tell him to not have an accident.