I think there is a chance that there could be a stable orbit such as this, but the orbital period of the secondary moon would have to be specific, so that the perturbations of the planet would cancel out over time.
I haven't done any calculations, but it might happen that the secondary moon's orbital period would have to be in resonance with the primary moon's orbital period...
More than likely this would actually lead to the moonlet leaving its parent body. Resonances tend to destabilize (there are gaps in the asteroid belt at resonances with Jupiter) more than they tend to stabilize (as with the orbits of Io, Europa, and Ganymede, which involves a resonance of three orbits, not two).
The biggest thing you would need would be for the planet and the primary moon to both have large hill spheres (which means both would have to be big and far out from the object they were orbiting). If the hill sphere of the planet is too small, then either the primary moon is close enough that tidal effects would strip the secondary moon out of orbit, or the primary moon is far enough out that it eventually will be stripped out of its orbit by the star and become a planet on its own.
If the hill sphere of the primary moon is too small, then it won't be able to have a moon of it's own for any significant length of time.
Think something like having the Earth/moon system orbit Neptune.
Neptune has two moons that are a third of an AU out (Semi Major Axis), but are quite small (one of them has quite an eccentric orbit, and reaches half an AU at apoposeidion. If you dropped an Earth-sized body into one of these orbits, it could certainly support its own moon. (Even a moon-sized body probably could).