Solute diffusion in solid polymers has tremendous applications in packaging, reservoir, and biomedical technologies but remains poorly understood. Diffusion of non-entangled linear solutes with chemically identical patterns (blobs) deviates dramatically in polymers in the solid-state (αlin > 1, Macromolecules 2013, 46, 874) from their behaviors in the molten state (αlin = 1, Macromolecules, 2007, 40, 3970). This work uses the scale invariance of the diffusivities, D, of linear probes D(N·Mblob + Manchor,T,Tg) = N-αlin(T,Tg)D(Mblob + Manchor,T,Tg) comprising N identical blobs of mass Mblob and possibly one different terminal pattern (anchor of mass Manchor) to evaluate the amounts of hole-free volume in seven polymers (aliphatic, semi-aromatic and aromatic) over a broad range of temperatures (-70 K ≤ T - Tg ≤ 160 K). The new parameterization of the concept of hole-free volumes opens the application of the free-volume theory (FVT) developed by Vrentas and Duda to practically any polymer, regardless of the availability of free-volume parameters. The quality of the estimations was tested with various probes including n-alkanes, 1-alcohols, n-alkyl acetates, and n-alkylbenzene. The effects of enthalpic and entropic effects of the blobs and the anchor were analyzed and quantified. Blind validation of the reformulated FVT was tested successfully by predicting from first principles the diffusivities of water and toluene in amorphous polyethylene terephthalate from 4 °C to 180 °C and in various other polymers. The new blob model would open the rational design of additives with controlled diffusivities in thermoplastics.