There doesn't seem to be such a word. Interstellar space within the Solar System is still just interstellar space.

There doesn't seem to be a demand for the word you're looking for either. We distinguish space by its contents; the space within the heliosphere is called the interplanetary medium (it contains solar plasma, dust, etc.), while the interstellar medium comprises sparse gaseous molecules from the Local Fluff and (if you go far enough) the rest of the Local Bubble.

Whereas there are about 10 oxygen molecules per cubic centimeter in the interplanetary medium, there is only 1 oxygen molecule per cubic centimeter in the interstellar medium. Seeing that there is no clear boundary for the Milky Way (nor is there any single definition for a boundary for galaxies), the intergalactic medium just vaguely describes the medium within galaxy filaments; here, the space is so rarified that there is only 1 oxygen molecule per cubic meter.