- Thermal conductivity (TC) enhancement of an insulating polymer matrix at low filler concentration is possible through the loading of a high aspect ratio, thermally conductive single filler. Unfortunately, the dispersion of high-aspect-ratio particles greatly influences the rheological behavior of the polymer host at relatively low volume fractions, which makes further polymer processing or mixing difficult. A possible remedy is using two (hybrid) fillers, differing in their aspect ratios: (1) a plate-like filler, which sharply increases both viscosity and TC, and (2) an isotropic filler, which gradually increases these properties. We examine this hypothesis in a thermosetting silicone rubber by loading it with different ratios, (1)/(2), of graphene nanoplatelets (GNPs) (1) and graphite powder (2). We constructed a “phase diagram” delineating two composite processability regions: solid-like (moldable) or fluid-like (pourable). This diagram may be employed to tailor the mixture’s viscosity to a desired TC value by varying the fillers’ volume fraction. The phase diagram highlights the low volume fraction value, above which the composite is solid-like (low processability) for a single high-aspect-ratio nanofiller. By using hybrid filling, one can overcome this limit and prepare a fluid-like composite at a desired TC, not accessible by the single nanofiller. Thus, it provides an indicative tool for polymer processing, especially in applications such as the encapsulation of electronic devices. This approach was demonstrated for a heat source (resistor) potted by silicon rubber graphene–graphite composites, for which a desired TC was obtained in both solid- and liquid-like regions.