I use a pick to play lead, because I find it hard to play fluidly with fingerstyle when I'm improvising, but I use mainly fingerstyle for rhythm. I can't stand hybrid picking... it feels really awkward to me, so when I switch from lead to rhythm, I put the pick between my lips in the corner of my mouth so it's easy to reach, but out of my hand.
For strumming, I usually don't use a pick. I can strum pretty well with just my hand by curling my fingers and kind of hitting the strings by flicking my wrist. I do a little bit of a flamenco rasguedo strum sometimes (though I suck at it), as well as a reverse strum (which isn't really a strum, I play ring, middle, index, thumb on four consecutive strings).
I've tried with hybrid picking and just don't like it. I keep an extra pick or two in my pocket in case I drop the pick, but I never have before...