I'm currently trying to play this cool jazz style with the help of cd's, videos and magazines such as guitarpart magazine.
Currently learning this piece. (he plays the complete piece at 4:57)

I understand that when picking a string, the first pick should always be downwards and then resting on the next string (similar to the apayando technique in classical guitar).

But since the righthand is adapting to that technique, it's further away from the bridge and can't automatically mute open strings like we do when using a normal righthand technique.

However, gypsy jazz players sound very clean to me.
How do they do it?

Also, I'm playing it on a normal acoustic guitar. Don't know if their guitars automatically have less unwanted noises.