The radii R of the area wetted by precursor films spreading from small droplets of polymer liquids are found to vary as R ~ t(ν) with ν = 0.31-0.42, in contradiction to the 0.5 exponent indicated in previous studies. The experiments reported here are for three of the same polymer liquids used in the previous studies-methyl-terminated polydimethylsiloxane (PDMS), tetrakis(2-ethylhexoxy)silane, and squalane-and a linear chain perfluoropolyether polymer. When the droplets dissipate into pancake-shaped films, the spreading kinetics slows down to a variety of different diffusion processes for the different molecules, as they are no longer forced to move en masse across the surface by molecules being fed into the center of the film by the droplet.