tapping power from a 3 way switch to an adjacent 2 way switch
Most likely, the hot wire is coming first to the other 3-way switch. From there, power comes on one of the two travelers to the 3-way you're working with. You've probably connected your new switch to one of the travelers. I'd bet if you switch the other 3-way, the light will come on and off together.

Without the hot coming to this 3-way, you cannot have independent operation of the new switch.