Where do you put your pick or how do you tuck it away quickly to go from strumming a verse or chorus of a song to doing a fingerpicking part.

Like for example something simple like fast car by Tracy chapman.

I don't like strumming with fingers don't sound right and just hurts and ruins my fingers.
With an electric you can slip it under the pickguard in many cases, acoustics don't have that option. I have a rubber pick holder on my mic stand for changing picks onstage, spares when I drop one, or when I don't need it because I finger pick a song. In a couple of cases I hold it in my lips when I need both in one song.

You can also get a spring loaded pick holder with a sticky thing on back that can be attached anywhere on a guitar but I hate to put anything sticky on my guitars.

If you're talking about onstage, check your local music stores, they should have a rubber pick holder with your name all over it. If you're talking about at home practicing, put it on a table or anywhere...

I would tend to think you're talking about onstage or band practice though, check these options out.


They also make them in necklace and bracelets I just found out...

Musicians Friend has the mic stand version for 4 bucks...

Hmmm...I wonder what this button does...
If you're at a show, many singer/guitarists have it tucked into the mic stand. I place it between my fingers.
There's a cut of Joe Pass on YouTube where he does this...Sticks the pick in his mouth. Of course, Joe isn't singing.....
Throw it on the ground. Stop playing and tell someone to give you your damn pick back.
In a single song, going from flat picking to finger picking is easy, the challenge is going from finger picking to flat picking...

I have seen players slide their pick up their hand with their thumb to free up their fingers for tapping, but this still immobilizes your thumb. I am not aware of any techniques where you can finger pick with a pick in your hand in order to switch back and forth easily, although there may be some I am not aware of...

Have you considered using hybrid picking for those parts?
Check out the Shark pick. It's a flat pick with an elastic band that holds it on your thumb, so you can go from flat-pick style strumming to finger style with a thumb pick.

Not a perfect solution but it works.
I've gotten pretty good at flipping between holding the pick with the index and thumb, then sliding it to my middle finger that curls a bit to hold it. I can only use my index and thumb at that point, but it helps in some situations, and is a fair trade-off between using just a pick, or only open hand fingerstyle.
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I have never done this and I'm hardly one to speak on any topic of advice, what about this.

attach a magnet to the side of your guitar, and use a metal pick?